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Cellular automata in operational probabilistic theories

Paolo Perinotti

QUIT Group, Dipartimento di Fisica, Università degli studi di Pavia, and INFN sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

The theory of cellular automata in oper-

ational probabilistic theories is developed.

We start introducing the composition of

inﬁnitely many elementary systems, and

then use this notion to deﬁne update rules

for such inﬁnite composite systems. The

notion of causal inﬂuence is introduced,

and its relation with the usual property of

signalling is discussed. We then introduce

homogeneity, namely the property of an

update rule to evolve every system in the

same way, and prove that systems evolving

by a homogeneous rule always correspond

to vertices of a Cayley graph. Next, we de-

ﬁne the notion of locality for update rules.

Cellular automata are then deﬁned as ho-

mogeneous and local update rules. Finally,

we prove a general version of the wrapping

lemma, that connects CA on diﬀerent Cay-

ley graphs sharing some small-scale struc-

ture of neighbourhoods.

Contents

1 Introduction 1

2 Detailed outlook 3

3 Operational Probabilistic Theories 4

3.1 Formal framework . . . . . . . . . 5

3.2 Causality and the no-restriction

hypothesis . . . . . . . . . . . . . . 8

3.3 Norms . . . . . . . . . . . . . . . . 10

4 The quasi-local algebra in OPTs 13

4.1 Quasi-local eﬀects . . . . . . . . . . 13

4.2 Extended states . . . . . . . . . . . 19

4.3 Quasi-local transformations . . . . 22

5 Global update rules 30

5.1 Update rule . . . . . . . . . . . . . 31

5.2 Admissibility and local action . . . 34

Paolo Perinotti: paolo.perinotti@unipv.it,

http://www.qubit.it

5.3 Causal inﬂuence . . . . . . . . . . . 36

5.3.1 Relation with signalling . . 38

5.4 Block decomposition . . . . . . . . 38

6 Homogeneity 40

7 Locality 47

8 Cellular Automata 50

8.1 Results . . . . . . . . . . . . . . . . 51

9 Examples 54

9.1 Classical case . . . . . . . . . . . . 54

9.2 Quantum case . . . . . . . . . . . . 55

9.3 Fermionic case . . . . . . . . . . . 55

10 Conclusion 56

Acknowledgments 56

A Identiﬁcation of the sup- and opera-

tional norm for eﬀects 56

B Quasi-local states 57

C Proof of identity 94 63

D Homogeneity and causal inﬂuence 64

E Proof of right and left invertibility of

a locally deﬁned GUR 64

F Proof of theorem 10 65

G Proof of lemmas 66 and 67 66

References 67

1 Introduction

In the last two decades a new approach to quan-

tum foundations arose, grounded on the quan-

tum information experience [1, 2, 3, 4]. This line

of research on the fundamental aspects of quan-

tum theory beneﬁts from new ideas, concepts and

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arXiv:1911.11216v3 [quant-ph] 6 Jul 2020

methods [5, 6, 7] that lead to a wealth of remark-

able results [8, 9, 10, 11]. In particular, quan-

tum theory can be now understood as a theory

of information processing, that is selected among

a universe of alternate theories [12, 13, 1, 14] by

operational principles about the possibility or im-

possibility to perform speciﬁc information pro-

cessing tasks [9, 15].

The scenario of alternate theories among which

the principles select quantum theory is called the

framework of Operational Probabilistic Theories

(OPTs) [8, 15], and has connections with the less

structured concept of Generalized Probabilistic

Theory (GPT) [5, 16, 17], as well as the dia-

grammatic category theoretical approach often

referred to as quantum picturialism [18, 19]. In-

spiration for the framework came also from quan-

tum logic [20, 21].

Quantum theory, namely the theory of Hilbert

spaces, density matrices, completely positive

maps and POVMs (in particular we refer to the

elegant exposition of Ref. [22]), can thus be re-

formulated as a special theory of information pro-

cessing. Besides the many advantages of this re-

sult, one has to face a main issue: the theory

as such is devoid of its physical content. El-

ementary systems are thought of as elementary

information carriers—brutally speaking, memory

cells—rather than elementary particles or ﬁelds in

space-time. While this framework is satisfactory

for an eﬀective, empirical description of physical

experiments, when it comes to provide a theo-

retical foundation for the physics of elementary

systems, the informational approach at this stage

calls for a way to re-embrace mechanical notions

such as mass, energy, position, space-time, and

complete the picture encompassing the dynamics

of quantum systems.

A recent proposal for this endeavour is based

on the idea that physical laws have to be ulti-

mately understood as algorithms, that make sys-

tems evolve, changing their state, exactly as the

memory cells of a computer are updated by the

run of an algorithm [23, 24, 25]. Such a program

already achieved successful results in the recon-

struction of Weyl’s, Dirac’s and Maxwell’s equa-

tions (for a comprehensive review see Ref. [26]).

The most natural candidate algorithm for de-

scribing a physical law in this context is a cellular

automaton. The theory of cellular automata is

a wide and established branch of computer sci-

ence. The notion of a cellular automaton for

quantum systems was ﬁrst devised as the quan-

tum version of its classical counterpart, e.g. in

Refs. [27, 28, 29, 30], but turned out to give rise to

a rather independent theory, developed starting

from Ref. [31]: the theory of Quantum Cellular

Automata (QCAs). The latter counts presently

various important results—see e.g. Refs. [32, 33],

just to mention a few. We stress that, most

commonly, QCAs are deﬁned to be reversible al-

gorithms, and most results in the literature are

proved with this hypothesis. It is known, how-

ever, that many desirable preoperties fail to hold

in the irreversible case (see e.g. Ref. [34]). In the

present work we will not consider irreversible cel-

lular automata, and leave this subject for further

studies.

In order to use cellular automata as candi-

date physical laws in the foundational perspec-

tive based on OPTs, one has two choices at

hand. The ﬁrst one is to start treating automata

within a deﬁnite theory, and this is the approach

adopted so far, in particular within Fermionic

theory [35, 36, 37]. In the linear case, Fermionic

cellular automata reduce to Quantum Walks, and

this brings in the picture all the tools from such

widely studied topic [38, 39, 40, 41, 42, 43]. The

non-linear case is far less studied [44, 45], and

does not oﬀer as many results for the analy-

sis. Needless to say, this approach faces diﬃcul-

ties that are speciﬁc of the theory at hand, and

prevents a comparison of diﬀerent theories on a

ground that is genuinely physical.

The second approach is initiated in the present

paper, and consists in deﬁning cellular automata

in the general context of OPTs. This perspective

oﬀers the possibility of extracting the essential

features of the theory of quantum or Fermionic

cellular automata, those that are not speciﬁc of

the theory but are well suited in any theory of

information processing. As a consequence, one

can generalise some results, and ﬁgure out why

and how others fail to extend to the broader sce-

nario. Moving a much less structured mathemat-

ical context, this approach can use only few tools,

but provides results that have the widest appli-

cability range.

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2 Detailed outlook

In this section we provide a short, non technical

discussion of the main results. The purpose of

this work is to deﬁne cellular automata in OPTs,

and prove some general results that will help ap-

plying the theory to special cases of interest.

A cellular automaton is an algorithm that up-

dates the information stored in an array of mem-

ory cells in discrete steps, in such a way that one

needs to read the content of a few neighbouring

cells to determine the state of a given cell at the

next step. Cellular automata in the literature

are often, but not always, deﬁned to be homoge-

neous: in this case the local rule for the update

of the cell is the same for every cell. Here we

will adopt homogeneity, but the subject is pre-

sented so that the generalisation of deﬁnitions to

inhomogeneous CA is straightforward.

Typically, the interesting case of a cellular au-

tomaton is the one involving an inﬁnite memory

array. The ﬁrst challenge we have to face is then

to extend the theory of OPTs from ﬁnite, arbi-

trarily large composite systems to actually inﬁ-

nite ones. This piece of theory has an interest

per se, for many reasons ranging from the possi-

bility to introduce thermodynamic limits to the

extension of the theory of C

∗

and von Neumann

algebras.

We then start with a review of the framework

of OPTs, and build the necessary tools to deﬁne

inﬁnite composite systems. The starting point

is the construction of mathematical objects that

describe measurements on ﬁnitely many systems

within an inﬁnite array— the OPT counterpart

of the space of local eﬀects of quantum theory.

Eﬀects for the inﬁnite system are then deﬁned as

limits of Cauchy sequences of local eﬀects. Since

the introduction of a suitable topology for the def-

inition of limits is needed, we open the paper with

section 3, where a review of OPTs is provided,

along with a few new results that will be useful,

and a rather consistent part of the section will be

dedicated to the introduction and discussion of

norms that will provide the necessary topological

framework for a consistent deﬁnition of limits.

The subject of the subsequent Section 4 is then

the construction of the Banach space of eﬀects for

the inﬁnite composite system, and consequently

the construction of the space of states as suitable

linear functionals on eﬀects. An important sub-

section will be dedicated to the construction of

the algebra of quasi-local transformations, that

allows for the description of transformations on

an inﬁnite system that can be arbitrarily well ap-

proximated by local operations on ﬁnitely many

subsystems. Indeed, an important part of the

theory of CAs in OPTs is built by ruling the way

in which quasi-local transformations are trans-

formed by the CA. In particular, the way in which

the CA propagates the eﬀects of a local transfor-

mation on surrounding subsystems will be the key

to the deﬁnition of the neighbourhood of the sub-

system, a concept that is central to the theory of

CAs.

Once this is done, the next step consists in

deﬁning update rules, and their admissibility con-

ditions, which are the subject of Section 5, along

with causal inﬂuence and a block-decomposition

theorem. Update rules are deﬁned in the ﬁrst

place as automorphisms V of the space of quasi-

local eﬀects, but an important request they must

abide is that when they act on a quasi-local trans-

formation A by conjugation as V A V

−1

, the ob-

tained transformation is again quasi-local.

The next step is taken in section 6, and con-

sists in deﬁning the property of homogeneity. The

latter presents with some diﬃculties, stemming

from the fact that, as we mentioned above, we

are deﬁning update rules prior to any mechanical

notion. This implies that even space-time is not

available at this fundamental stage, and without

geometry we cannot deﬁne homogeneity as trans-

lational invariance under the group correspond-

ing to a given space-time. On the contrary, we

will deﬁne homogeneity by formalising the idea

that every single cell has to be treated equally

by the update rule, and this notion will be de-

ﬁned operationally, requiring that no experiment

made of local operations will allow establishing

any diﬀerence between cells. As a consequence

of homogeneity, one can prove that every homo-

geneous update rule is underpinned by the Cay-

ley graph of some group, generalising a result of

Refs. [25, 46]. The mathematical theory known as

geometric group theory then tells us that the Cay-

ley graph representing the causal connections of

cells in the memory array, being a metric space,

uniquely identiﬁes an equivalence class of met-

ric spaces, that captures both the algebraic and

geometric essential features of the group. Aston-

ishingly, for Cayley graphs of ﬁnitely presented

groups—which is the case for cellular automata

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as we deﬁne them here—the equivalence class al-

ways contains a smooth manifold of dimension at

most four (see Ref. [47], pag. 90). This result

means that we can always think of a cellular au-

tomaton as if it was embedded in a Riemannian

manifold, in such a way that the Riemannian dis-

tance between nodes is almost the same as the

distance between nodes given in terms of steps

along the graph edges.

Also the notion of locality, presented in sec-

tion 7, comes with its own diﬃculties, that can be

overcome by proving a generalisation of the result

known as “unitarity plus causality implies local-

izability" [32]. This is where the theory, which is

inspired by that of QCAs, deeply diﬀers from the

theory of classical cellular automata. In particu-

lar, considering the collection of transformations

V A V

−1

for all A acting on a given system g, we

will deﬁne the neighbourhood of g as the set of

systems on which transformations V A V

−1

act

non trivially.

Once the theory of cellular automata is fully

developed, some results are proved in Section 8.

In particular, a very useful theorem is the wrap-

ping lemma, which under very wide hypotheses—

though not universal—allows for the classiﬁcation

of automata on a given inﬁnite graph by classi-

fying automata on any suitably “wrapped" ﬁnite

version of the same graph.

In Section 9, a few examples are reviewed. In

particular, using the general notion of locality,

we apply the deﬁnition of causal inﬂuence and

the neighbourhood scheme to the case of classical

cellular automata. With the above deﬁnitions at

hand, we show that allegedly local automata are

actually non-local. This solves a long standing

puzzle about the connection between classical and

quantum automata—the so-called quantisation of

classical automata [31, 48].

The paper is concluded by Section 10 with a

summary of the results and some closing remarks.

3 Operational Probabilistic Theories

In this section we will provide a thorough intro-

duction to OPTs, which is partly a review of the

literature, and partly presentation of new results

that will be used in the remainder. We provide

here a sketchy introduction before going through

formal deﬁnitions, and use Quantum Theory as

an illustrative example for the main notions in

the framework.

Quantum theory is about system types A

1

—namely complex Hilbert spaces H

A

that are

classiﬁed by their dimension d

A

= dim(H

A

)—and

transformations that can occur on systems as a

consequence of undergoing a test. Tests are rep-

resented by quantum instruments E

X

= {E

i

}

i∈X

,

i.e. a collection of Completely Positive (CP) maps

E

i

that sum to a trace-preserving one (a channel):

E =

P

i∈X

E

i

. The quantum operation E

i

—a CP

trace non-increasing map—represents the change

in the system occurring upon the event of a spe-

ciﬁc outcome i ∈ X in the test. In the diagram-

matic language of OPTs systems are represented

as labelled wires, and instruments or quantum op-

erations as boxes with an input and output wire,

e.g.

A

E

X

B

,

A

E

i

B

,

respecitvely. States can be considered as special

transformations where the input system is I, hav-

ing H

I

= C, i.e. d

I

= 1. Notice that the a prepa-

ration test, i.e. the most general test from I to A,

represents a probabilistic preparation procedure

where some state in the collection P

X

= {ρ

i

}

i∈X

is prepared, with the sub-normalised density ma-

trix ρ

i

occurring upon reading the outcome i ∈ X.

The probability of occurrence of ρ

i

is Tr[ρ

i

], and

ρ =

P

i∈X

ρ

i

has unit trace. Preparation tests

or states for system A are represented by special

diagrams

P

X

A

,

ρ

i

A

,

respectively. The space H

A

of Hermitian opera-

tors on H

A

, whose dimension is D

A

= d

2

A

, is the

real span of states of system A.

A POVM Q

Y

= {Q

j

}

j∈Y

is a collection of ef-

fects 0 ≤ Q

j

≤ I

A

that sum to the identity opera-

tor:

P

j∈Y

Q

j

= I

A

. This kind of test can be seen

as a quantum instrument from A to I, namely a

collection of linear functionals on the real space

spanned by density matrices, where the function-

als a

j

(ρ) are deﬁned as a

j

(ρ)

:

= Tr[ρQ

j

]. The

1

We remark that for the purpose of information pro-

cessing the type of a system is captured by the dimension

of the corresponding Hilbert space, e.g. the electron spin

is of the same type as the photon polarisation. This is

slightly diﬀerent than the usual notion of system type in

physics, where the two types in the above example are

diﬀerent.

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diagrammatic representation of POVMs and ef-

fects is the following

A

Q

Y

,

A

Q

j

,

respectively.

Composite systems, such as AB, correspond to

to the Hilbert space H

A

⊗H

B

of dimension d

AB

=

d

A

d

B

. Tests {E

i

}

i∈X

from A to B and {F

j

}

j∈Y

from B to C can be run in sequence, obtaining a

new test: {D

ij

}

(i,j)∈X×Y

where D

ij

:

= F

j

E

i

. On

the other hand, tests {E

i

}

i∈X

from A to B and

{E

j

}

j∈Y

from B to B

0

on subsystems of a compos-

ite system AA

0

can be run in parallel, obtaining

the test {D

ij

}

(i,j)∈X×Y

with D

ij

:

= E

i

⊗ F

j

. In

diagrams, we draw for quantum operations

A

F

j

E

i

C

=

A

E

i

B

F

j

C

,

AA

0

E

i

⊗ F

j

BB

0

=

A

E

i

B

A

0

F

j

B

0

,

and analogously for instruments.

Very relevant structures in the theory are the

real space H

A

of Hermitian operators on H

A

,

spanned by quantum states, the cone of positive

operators in P

A

⊆ H

A

, the convex set of states

[[A]], obtained by intersecting the positive cone

with the half-space Tr[X] ≤ 1, and the convex

set of deterministic states [[A]]

1

, obtained by in-

tersecting the positive cone with the aﬃne hy-

perplane Tr[X] = 1. Similar structures can be

generalised to the space spanned by quantum op-

erations. In the general framework of OPTs, we

will systematically refer to the generalisation of

the above concepts.

3.1 Formal framework

The framework of OPTs is meant to capture the

main traits of Quantum Theory (shared e.g. by

Classical Theory, or Fermionic Theory, etc.) sum-

marised above, and use them as deﬁning proper-

ties of a family of abstract theories that might

be candidates for an alternative representation of

elementary physical systems and their transfor-

mations. In the remainder of this section we pro-

vide a brief review of the framework of OPTs (for

reference see e.g. [15, 8, 9]). Some of the most

relevant diﬀerences with respect to the quantum

case stem from the fact that systems might not

compose with the tensor product rule.

We warn the reader that this presentation has

some elements that are slightly diﬀerent from

other reviews in the literature, and is tailored

to ease the presentation of subsequent material.

Most of the results in the present sections are

new, and their proof will be given. Those the-

orems that are not original are not proved, and

due reference is provided.

An operational theory Θ consists in i) a col-

lection Test(Θ) of tests T

A→B

X

, each labelled by

input and output letters from a collection Sys(Θ)

denoting system types, e.g. A → B—that will

be systematically omitted—and by a ﬁnite set of

outcomes X; for every pair of types A, B ∈ Sys(Θ)

the set of tests of type A → B is denoted hhA →

Bii; ii) a very basic associative rule for sequential

composition: the test T

X

∈ hhA → Bii can be fol-

lowed by the test S

Y

∈ hhA

0

→ B

0

ii if A

0

≡ B, thus

obtaining the sequential composition ST

X×Y

∈

hhA → B

0

ii; iii) a rule ⊗ : (A, B) 7→ AB for com-

posing labels in parallel, and a corresponding rule

for tests ⊗ : (S

X

, T

Y

) 7→ (S ⊗ T)

X×Y

, with the fol-

lowing properties

1. Associativity: (AB)C = A(BC).

2. For every S

X

∈ hhA → Bii and T

Y

∈ hhC →

Dii, one has S

X

⊗ T

Y

∈ hhAC → BDii. Asso-

ciativity of ⊗ holds:

(S

X

⊗ T

Y

) ⊗ W

Z

= S

X

⊗ (T

Y

⊗ W

Z

).

3. Unit: there is a label I such that IA = AI =

A for every A ∈ Sys(Θ).

4. Identity: for every A ∈ Sys(Θ), a test I

A

∈

hhA → Aii such that I

B

S

X

= S

X

I

A

= S

X

, for

every S

X

∈ hhA → Bii.

5. For every A

X

∈ hhA → Bii, B

Y

∈ hhB → Cii,

D

Z

∈ hhD → Eii, E

W

∈ hhE → Fii, one has

(B

Y

⊗ E

W

)(A

X

⊗ D

Z

) = (B

Y

A

X

) ⊗ (E

W

D

Z

).

(1)

6. Braiding: for every pair of system types A, B,

there exist tests S

AB

, S

∗

AB

∈ hhAB → BAii

such that S

∗

BA

S

AB

= S

BA

S

∗

AB

= I

AB

, and

S

AB

(A

X

⊗ B

Y

) = (B

Y

⊗ A

X

)S

AB

. Moreover,

S

(AB)C

= (I

A

⊗ S

BC

)(S

AC

⊗ I

B

),

S

A(BC)

= (S

AB

⊗ I

C

)(I

B

⊗ S

AC

).

When S

∗

AB

≡ S

AB

, the theory is symmetric.

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All the theories developed so far are symmetric.

All tests of an operational theory are (ﬁnite)

collections of events: hhA → Bii 3 R

X

= {R

i

}

i∈X

.

If hhA → Bii 3 R

X

= {R

i

}

i∈X

and hhB → Cii 3

T

Y

= {T

j

}

j∈Y

, then

hhA → Cii 3 (TR)

X×Y

:

= {T

j

R

i

}

(i,j)∈X×Y

.

Similarly, for R

X

∈ hhA → Bii and T

Y

∈ hhC →

Dii,

(R ⊗ T)

X×Y

:

= {R

i

⊗ T

j

}

(i,j)∈X×Y

.

The set of events of tests in hhA → Bii is de-

noted by [[A → B]]. By the properties of se-

quential and parallel composition of tests, one

can easily derive associativity of sequential and

parallel composition of events, as well as the ana-

logue of Eq. (1). For every test T

X

∈ hhA → Bii

with T

X

= {T

i

}

i∈X

, and every disjoint partition

{X

j

}

j∈Y

of X =

S

j∈Y

X

j

, one has a coarse grain-

ing operation that maps T

X

to T

0

Y

∈ hhA → Bii,

with T

0

Y

= {T

0

j

}

j∈Y

. We deﬁne T

X

j

:

= T

0

j

. The

parallel and sequential compositions distribute

over coarse graining:

T

X

j

⊗ R

k

= (T ⊗ R)

X

j

×{k}

,

A

l

T

X

j

B

k

= (A T B)

{l}×X

j

×{k}

.

Notice that for every test T

X

∈ hhA → Bii there

exists the singleton test T

0

∗

:

= {T

X

}. One can

easily prove that the identity test I

A

is a sin-

gleton: I

A

= {I

A

}, and I

B

T = T I

A

for

every event T ∈ [[A → B]]. Similarly, for

S

AB

= {S

AB

} and S

∗

AB

= {S

∗

AB

} we have

S

∗

BA

S

AB

= S

BA

S

∗

AB

= I

AB

. The collection of

events of an operational theory Θ will be denoted

by Ev(Θ). The above requirements make the col-

lections Test(Θ) and Ev(Θ) the families of mor-

phisms of two braided monoidal categories with

the same objects—system types Sys(Θ).

An operational theory is an OPT if the tests

hhI → Iii are probability distributions: 1 ≥ T

i

=

p

i

≥ 0, so that

P

i∈X

p

i

= 1, and given two tests

S

X

, T

Y

∈ hhI → Iii with S

i

= p

i

and T

i

= q

i

, the

following identities hold

S

i

⊗ T

j

= S

i

T

j

:

= p

i

q

j

,

T

X

j

:

=

X

i∈X

j

p

i

,

meaning that events in the same test are mutually

exclusive and events in diﬀerent tests of system I

are independent. While it is immediate that 1 ∈

[[I → I]]—since {1}

∗

= {I

I

} is the only singleton

test—we will assume that 0 ∈ [[I → I]]. This

means that we can consider e.g. tests of the form

{1, 0, 0}.

Events in [[A → B]] are called transformations.

As a consequence of the above deﬁnitions, every

set [[A]]

:

= [[I → A]] can be viewed as a set of

functionals on [[

¯

A]]. As such, it can be viewed as

a spanning subset of the real vector space [[A]]

R

of linear funcitonals on [[

¯

A]]. On the other hand

[[

¯

A]]

:

= [[A → I]] is a separating set of positive

linear functionals on [[A]], which then spans the

dual space [[A]]

∗

R

=: [[

¯

A]]

R

. The dimension D

A

of

[[A]]

R

(which is the same as that of [[

¯

A]]

R

) is called

size of system A. One can easily prove that in

any OPT Θ, I is the unique system with unit size

D

I

= 1. Using the properties of parallel compo-

sition, one can also prove that D

AB

≥ D

A

D

B

.

Events in [[A]] are called states, and denoted by

lower-case greek letters, e.g. ρ, while events in [[

¯

A]]

are called eﬀects, and denoted by lower-case latin

letters, e.g. a. When it is appropriate, we will

use the symbol |ρ) to denote a state, and (a| to

denote an eﬀect. We will also use the circuit no-

tation, where we denote states, transformations

and eﬀects by the symbols

ρ

A

,

A

B

,

A

a

,

respectively. Sequential composition of A ∈

[[A → B]] and B ∈ [[B → C]] is denoted by the

diagram

A

BA

C

=

A

B

C

.

For composite systems we use diagrams with mul-

tiple wires, e.g.

A

B

C D

.

The identity will be omitted:

A

I

A

=

A

. The swap S

AB

and its inverse will be de-

noted as follows

A

S

B

B A

=

B

A

S

∗

B

B A

=

B

A

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In the present paper we will always assume

that the theory under consideration is symmet-

ric, however all the results will be straightfor-

wardly generalisable. We will consequently draw

the swap S

AB

as

A

S

B

B A

=

A

S

∗

B

B A

=

B

A

An OPT Θ is speciﬁed by the collections of

systems and tests, along with the parallel com-

position rule ⊗

Θ ≡ (Test(Θ), Sys(Θ), ⊗).

Deﬁnition 1 (Equal transformations). Let

A , B ∈ [[A → B]]. Then we deﬁne A = B

if for every system C and every ρ ∈ [[AC]] and

a ∈ [[

¯

B

¯

C]], one has

ρ

A

B

a

C

=

ρ

A

B

a

C

Notice that, since states separate eﬀects and

viceversa eﬀects separate states, the above deﬁ-

nition is equivalent to the two following equality

criteria.

Lemma 1. Let A , B ∈ [[A → B]]. Then the

following conditions are equivalent.

1. A = B.

2. For every C and every ρ ∈ [[AC]], one has

ρ

A

B

C

=

ρ

A

B

C

.

(2)

3. For every C and every a ∈ [[

¯

B

¯

C]], one has

A

B

a

C

=

A

B

a

C

.

(3)

One can show [8] that events T ∈ [[A → B]]

can be identiﬁed with a family of linear maps,

one for every C, that characterize the action of

T ⊗ I

C

on [[AC]]

R

as a linear map to [[BC]]

R

.

As anticipated in the introductory paragraph, we

remind the reader that some of the diﬃculties

that we will be faced with in the remainder orig-

inate from the fact that in a general theory it is

not true that [[AB]]

R

= [[A]]

R

⊗ [[B]]

R

, but only

[[A]]

R

⊗[[B]]

R

⊆ [[AB]]

R

. As a consequence, the lin-

ear map representing T on [[A]]

R

is not suﬃcient

to determine the linear map representing T ⊗I

C

on [[AC]]

R

.

One can easily prove that [[A → B]] spans a real

vector space [[A → B]]

R

. Being [[A → B]] spanning

for [[A → B]]

R

, the criteria of deﬁnition 1 and

lemma 1 hold for A , B ∈ [[A → B]]

R

. Elements

of [[A → B]]

R

are called generalized events. Every

space [[A → B]]

R

has a zero event 0

A→B

:

= 0

I→I

⊗

T ∈ [[A → B]], where T is an arbitrary event in

[[A → B]]. As a consequence of the coarse graining

rule for tests on [[I → I]], one can easily show that

coarse graining of two or more transformations is

represented by their sum. Precisely, given a test

{T

i

}

i∈X

⊆ [[A → B]], for X

0

= {0, 1} ⊆ X one has

T

0

= T

0

+ T

1

, namely for every system C it is

T

0

⊗ I

C

= T

0

⊗ I

C

+ T

1

⊗ I

C

. Finally, every

set [[A → B]] has a subset consisting in singleton

events, that we denote by [[A → B]]

1

, and call

deterministic. For A, B 6= I, a deterministic event

is called channel. A channel U ∈ [[A → B]]

1

is

reversible if there exists a channel V ∈ [[B → A]]

1

such that V U = I

A

, U V = I

B

.

Let us now deﬁne the cones

[[A → B]]

+

:

= {λT | λ ≥ 0, T ∈ [[A → B]]}.

We will often write A  0 as a shorthand for

A ∈ [[A → B]]

+

. The cone [[A → B]]

+

introduces

a partial ordering in [[A → B]]

R

, deﬁned by

A  B ⇔ (A − B)  0.

OPTs are assumed to have all sets [[A → B]] (and

thus also cones [[A → B]]

+

) closed in the opera-

tional norm.

Two systems may be operationally equivalent

if they can be mapped one onto the other via a

reversible transformation. Clearly, in this case

every processing of the ﬁrst system is perfectly

simulated by a processing of the other, and vicev-

ersa.We deﬁne operationally equivalent systems

as follows.

Deﬁnition 2. Let A, B be two systems. We say

that A and B are operationally equivalent, in for-

mula A

∼

=

B, if there exists a reversible transfor-

mation U ∈ [[A → B]]

1

.

Deﬁnition 3. If A

1

and A

2

are operationally

equivalent through U and B

1

and B

2

through V ,

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then, for every system C, A

1

∈ [[A

1

C → B

1

C]]

R

and A

2

∈ [[A

2

C → B

2

C]]

R

are operationally

equivalent if A

2

= (V ⊗ I

C

)A

1

(U

−1

⊗ I

C

). In

particular, ρ

1

∈ [[A

1

C]]

R

and ρ

2

∈ [[A

2

C]]

R

are

operationally equivalent if ρ

2

= (U ⊗I

C

)ρ

1

, and

a

1

∈ [[

¯

A

1

¯

C]]

R

and a

2

∈ [[

¯

A

2

¯

C]]

R

are operationally

equivalent if a

2

= a

1

(U

−1

⊗ I

C

).

Lemma 2. Let AB be operationally equivalent to

A. Then B must be the trivial system I.

Proof. Since D

AB

≥ D

A

D

B

≥ D

A

, if AB

∼

=

A

the chain of inequalities is saturated, and thus

D

B

= 1.

3.2 Causality and the no-restriction hypothesis

In the remainder of the paper we will focus on

causal theories. The causality property, that

we deﬁne right away, characterizes those theories

where signals can propagate only in the direc-

tion deﬁned by input and output of processes,

within a cone of causal inﬂuence determined by

interactions between systems. These are the only

theories where one can consistently use informa-

tion acquired in a set of tests to condition the

choice of subsequent tests, where the partial or-

dering we are referring to is that determined by

the input/output direction.

Deﬁnition 4 (Causal theories). A theory (T , A)

is causal if for every test {T

i

}

i∈X

and every col-

lection of tests {S

i

j

}

j∈Y

labelled by i ∈ Y, the

generalized test {C

i,j

}

(i,j)∈X×Y

with

A

C

i,j

C

:

=

A

T

i

B

S

i

j

C

,

is a test of the theory.

Notice that this notion of causality is strictly

stronger than the one usually adopted in the lit-

erature about OPTs, that can be summarised

as uniqueness of the deterministic eﬀect (see

e.g. Refs. [8, 9, 15]). Indeed, a ﬁrst result of

crucial importance derives uniqueness of the de-

terministic eﬀect for every system type in causal

theories.

Theorem 1 ([8, 15]). In a causal theory, for ev-

ery system type A the set of deterministic eﬀects

[[

¯

A]]

1

is the singleton {e

A

}.

A proof of the above theorem, that proceeds

by contradiction, can be found in the mentioned

references. We remark that, being the notion of

causality in these references diﬀerent form the one

deﬁned here, the theorem has a slightly diﬀerent

statement. In the same references, one can ﬁnd

the equivalence of uniqueness of the deterministic

eﬀect with non-signalling from output to input.

Theorem 2 ([8, 15]). In an OPT, every system

type A has a unique deterministic eﬀect if and

only if the marginal probabilities of preparation

tests cannot depend on the choice of subsequent

observation tests.

As a consequence of our notion of causality, in

a non-deterministic theory (i.e. a theory where

[[I]] 6= {0, 1}), every convex combination of tests

is a test (see e.g. [15]). This implies that the sets

[[A]]

∗

, [[

¯

A]]

∗

, [[A → B]]

∗

are convex, where ∗ can be

replaced by nothing, 1 or +. Moreover, the sets

with ∗ = + are convex cones.

Another important consequence of causality is

that a transformation C ∈ [[A → B]] is determin-

istic if and only if it maps the deterministic eﬀect

e

B

to the deterministic eﬀect e

A

.

Theorem 3 ([8, 15]). In a causal theory, a trans-

formation C ∈ [[A → B]] is a channel iﬀ

A

C

B

e

=

A

e

.

In causal theories, one can always assume that

every state is proportional to a deterministic one.

Indeed, including in the theory every state ρ ∈

[[A]]

+

such that (e

A

|ρ) = 1 does not introduce any

inconsistency with the set of transformations, as

we now prove. Let Θ be a causal OPT, and deﬁne

the theory Θ

0

through the bijection κ : Sys(Θ) →

Sys(Θ

0

) :: A 7→ A

0

, with

[[A

0

→ B

0

]]

R

≡ [[A → B]]

R

,

[[A

0

→ B

0

]]

1

:

= {T  0 | (e

B

|T = (e

A

|},

[[A

0

→ B

0

]]

:

= {T  0 | ∃C ∈ [[A

0

→ B

0

]]

1

, C  T },

hhA

0

→ B

0

ii

:

= {T ⊆ [[A

0

→ B

0

]] |

X

T ∈T

T ∈ [[A

0

→ B

0

]]

1

}.

where  denotes the ordering induced by the cones

of the theory Θ, and the cardinality of sets T

in the last deﬁnition is implicitly assumed to be

ﬁnite.

Theorem 4. Let Θ be a causal OPT, and con-

sider the theory Θ

0

deﬁned above. Then Θ

0

is an

OPT with a unique deterministic eﬀect e

A

0

for

every A

0

.

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Proof. All the compositional structures of Θ are

inherited by Θ

0

. The deﬁning conditions in the

special case of [[A

0

]] = [[I

0

→ A

0

]] give

[[A

0

]] = {ρ ∈ [[A]]

+

| (e

A

|ρ) ≤ 1},

while for [[

¯

A

0

]] = [[A

0

→ I

0

]] they give

[[

¯

A

0

]]

1

= {e

A

}.

What remains to be proved is that the set of

transformations is closed under sequential and

parallel composition. First of all, we observe that

by theorem 3 [[A → B]]

1

⊆ [[A

0

→ B

0

]]

1

, thus

[[A → B]] ⊆ [[A

0

→ B

0

]]. This makes [[A → B]]

+

⊆

[[A

0

→ B

0

]]

+

. On the other hand, since by def-

inition it is also [[A

0

→ B

0

]] ⊆ [[A → B]]

+

, we

have [[A

0

→ B

0

]]

+

⊆ [[A → B]]

+

, which makes

[[A

0

→ B

0

]]

+

≡ [[A → B]]

+

. Thus the ordering  is

the same for the two theories. As a consequence,

the preservation of cones under the two composi-

tions is inherited by Θ

0

from the same property in

Θ. Now, if A ∈ [[A

0

→ B

0

]]

1

and B ∈ [[B

0

→ C

0

]]

1

,

then

(e

C

|BA = (e

B

|A = (e

A

|,

thus BA ∈ [[A

0

→ C

0

]]

1

. Moreover, thanks to

causality of the theory Θ one has e

AB

= e

A

⊗ e

B

,

thus for A ∈ [[A

0

→ B

0

]]

1

and B ∈ [[C

0

→ D

0

]]

1

,

(e

BD

|A ⊗ B = (e

B

|A ⊗ (e

D

|B

= (e

A

| ⊗ (e

C

| = (e

AC

|,

and consequently A ⊗ B ∈ [[A

0

C

0

→ B

0

D

0

]]

1

.

Now, given events A , B in the theory Θ

0

, by deﬁ-

nition we have C , D deterministic in Θ

0

such that

C  A and D  B, thus

F

:

=(D − B)(C − A ) + (D − B)A

+ B(C − A )  0,

G

:

=(C − A ) ⊗ (D − B)

+ (C − A ) ⊗ B + A ⊗ (D − B)  0.

Finally, F + BA  BA and G + A ⊗ B  A ⊗

B are channels, and thus BA and A ⊗ B are

events.

One can now show that the new theory Θ

0

is

causal.

Corollary 1. Let Θ be a causal OPT, and Θ

0

as

in theorem 4. Then Θ

0

is a causal OPT.

Proof. Let {T

i

}

i∈X

be a test in [[A

0

→ B

0

]], and

{S

i

j

}

j∈Y

be tests in [[B

0

→ C

0

]] for every i ∈ X.

Then

∀i ∈ X

X

j∈Y

(e

C

0

|S

i

j

= (e

B

0

|,

X

i∈X

(e

B

0

|T

i

= (e

A

0

|.

This implies that

X

i∈X

X

j∈Y

(e

C

0

|W

i,j

= (e

A

0

|,

W

i,j

:

= S

i

j

T

i

∈ [[A

0

→ C

0

]] ∀i ∈ X, j ∈ Y.

As a consequence, every conditional test W

i,j

:

=

S

i

j

T

i

∈ [[A

0

→ C

0

]] is admitted in the theory

Θ

0

.

Remark 1. In the remainder, we will focus on

theories satisfying causality and the further re-

quirements

[[A → B]]

1

= {T  0 | (e

B

|T = (e

A

|},

[[A → B]] = {T  0 | ∃C ∈ [[A → B]]

1

, C  T },

hhA → Bii = {T ⊆ [[A → B]] |

X

T ∈T

T ∈ [[A → B]]

1

}.

In particular, this implies that [[A]] = {ρ ∈ [[A]]

+

|

(e

A

|ρ) ≤ 1}. Thanks to corollary 1, this is not a

signiﬁcant restriction.

We now narrow down focus on theories that

satisfy the no-restriction hypothesis. Let us con-

sider the set of preparation-tests for every system

of the theory Θ. Then we will complete the set

of tests allowing for all those tests that transform

preparation tests to preparation tests, even when

applied locally. This requirement is the general-

isation of the assumption made in quantum the-

ory that a map is a transformation if and only

if it is completely positive and trace non increas-

ing, and a collection of transformations is a test

if and only if the sum of its elements is a channel,

i.e. completely positive and trace-preserving.

Assumption 1. Let A ∈ [[A → B]]

R

. The no-

restriction hypothesis consists in the requirement

that T ∈ hhA → Bii if and only if for every C and

every P ∈ hhI → ACii, one has (T ⊗ I

C

)P ∈ hhI →

BCii.

We will write A  0 if for every C and every

P ∈ [[AC]]

+

, one has (A ⊗ I

C

)P ∈ [[BC]]

+

. The

set of transformations A  0 is a cone, that we

denote by

K(A → B)

:

= {A ∈ [[A → B]]

R

| A  0}.

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The above deﬁned cone introduces a (partial) or-

dering in [[A → B]]

R

, deﬁned by

A  B ⇔ A − B  0. (4)

Notice that, under the no-restriction hypothe-

sis, the following identity holds

K(A → B) = [[A → B]]

+

,

and thus for every A , B ∈ [[A → B]]

R

,

A  B ⇔ A  B. (5)

We remark that, under assumption 1, the hy-

pothesis of theorem 3 can be relaxed to C ∈

[[A → B]]

+

, since the latter implies that C = λC

0

for some C

0

∈ [[A → B]] and λ ≥ 0. Then,

C  0. Moreover, since C sends e

B

to e

A

,

one has (C ⊗ I

C

)[[AC]]

1

⊆ [[BC]]

1

, and thus

(C ⊗ I

C

)[[AC]] ⊆ [[BC]]. By the no-restriction

hypothesis, this is tantamount to C ∈ [[A → B]].

We can now prove a theorem that is a very

important consequence of the no-restriction hy-

pothesis, along with causality.

Theorem 5. In a theory satisfying the no-

restriction hypothesis, let T ∈ [[A → B]]

R

. Then

T , C − T  0 for some channel C ∈ [[A → B]]

1

iﬀ T ∈ [[A → B]].

Proof. The hypothesis T , C − T  0 is equiva-

lent to T , C − T ∈ [[A → B]]

+

. Equivalently,

for every C and every P ∈ [[AC]]

+

, one has

(X ⊗ I

C

)P ∈ [[BC]]

+

for X = T , C − T . Let

now P ∈ [[AC]]. Then one has

(e

BC

|[(C − T ) ⊗ I

C

]|P ) ≥ 0,

and thus

0 ≤ (e

BC

|(T ⊗ I

C

)|P ) ≤ (e

AC

|P ) ≤ 1.

This implies that (T ⊗ I

C

)P ∈ [[AC]]. Finally,

by the no-restriction hypothesis, T ∈ [[A → B]].

The converse statement is trivial.

In the literature [10, 11] one can often ﬁnd a dif-

ferent requirement under the name “no-restriction

hypothesis”, namely that for every system A the

set [[

¯

A]] coincides with the set of functionals a

on [[A]]

R

that satisfy 0 ≤ (a|ρ) ≤ 1 for ev-

ery ρ ∈ [[A]]. The last condition may be nei-

ther necessary nor suﬃcient for the no-restriction

hypothesis as we state it, despite counterexam-

ples are still unknown for both cases. Indeed,

the no-restriction hypothesis for eﬀects imposes

the requirement that for every C and every state

P ∈ [[AC]], one has

P

A

a

C

∈ [[C]] . (6)

3.3 Norms

As the ﬁrst step in the present work is to con-

struct inﬁnite composite systems, we will need a

thorough notion of sequences and limits. From a

topological point of view the vector spaces that

we constructed so far have no special structure,

however the operational procedures for discrim-

ination of processes provide a natural deﬁnition

of distance between events. Such a distance is re-

lated to the success probability in discriminating

events. Making the space of events into a metric

space, the operational distance immediately pro-

vides a topological structure through the induced

operational norm. However, in order to make the

space of events of inﬁnite composite systems into

a Banach algebra, a stronger norm is needed, that

is introduced here: the sup-norm. In the quan-

tum and classical case the two norms coincide.

The operational norm k·k

op

for states is deﬁned

as follows [15].

Deﬁnition 5. The operational norm on [[A]]

R

is

kρk

op

:

= sup

a∈[[

¯

A]]

(2a − e|ρ) = sup

a

0

,a

1

∈[[

¯

A]]

a

0

+a

1

=e

A

(a

0

− a

1

|ρ).

(7)

The operational norm k·k

op

for transformations

A ∈ [[A → B]]

R

on ﬁnite systems is then deﬁned

as

kA k

op

:

= sup

C,Ψ∈[[AC]]

1

k(A ⊗ I

C

)Ψk

op

= sup

C,Ψ∈[[AC]]

sup

a∈[[

¯

B

¯

C]]

(2a − e|

˜

A ⊗ I

C

|Ψ).

In the special case of eﬀects a ∈ [[

¯

A]]

R

we have

kak

op

:

= sup

C,Ψ∈[[AC]]

k(a ⊗ I

C

)Ψk

op

= sup

C,Ψ∈[[AC]]

sup

b∈[[

¯

C]]

(a ⊗ [2b − e

C

]|Ψ).

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As a consequence,

kak

op

= sup

ρ

0

,ρ

1

∈[[A]]

ρ

0

+ρ

1

∈[[A]]

(a|ρ

0

− ρ

1

)

= sup

p∈[0,1]

{p sup

ρ∈[[A]]

(a|ρ) − (1 − p) inf

σ∈[[A]]

(a|σ)}

= max{ sup

ρ∈[[A]]

(a|ρ), inf

σ∈[[A]]

(a|σ)}

= sup

ρ∈[[A]]

|(a|ρ)|.

Here we only prove one new result about the op-

erational norm, that we will use in the following.

Lemma 3. Let ρ ∈ [[A]]

R

and σ ∈ [[B]]

1

. Then

kρ ⊗ σk

op

= kρk

op

. (8)

Proof. By deﬁnition it is

kρ ⊗ σk

op

= sup

(a

0

,a

1

)

(a

0

− a

1

|ρ ⊗ σ)

= sup

(b

0

,b

1

)∈M

(b

0

− b

1

|ρ),

where M

:

= {(b

0

, b

1

) | b

i

= a

i

(I

A

⊗ σ)}. Thus

kρ ⊗ σk

op

≤ kρk

op

.

On the other hand, for every binary observation-

test (a

0

, a

1

) in [[

¯

A]] one has

a

i

= (a

i

⊗ e

B

)(I

A

⊗ σ),

and thus M actually contains every possible bi-

nary observation-test on A. Finally, this implies

that

kρ ⊗ σk

op

= kρk

op

.

For more details on k·k

op

see Refs. [8, 15].

We now proceed to deﬁne the sup-norm.

Deﬁnition 6. The sup-norm kA k

sup

of A ∈

[[A → B]]

R

is deﬁned as

kA k

sup

:

= inf J(A ),

J(B)

:

= {λ | ∃C ∈ [[A → B]]

1

, λC  B  −λC }.

We now show that k·k

sup

actually deﬁnes a

norm.

Proposition 1. The sup-norm on [[A → B]]

R

is

well deﬁned:

1. kA k

sup

is non-negative, and it is null iﬀ

A = 0,

2. for µ ∈ R one has kµA k

sup

= |µ|kA k

sup

,

3. kA + Bk

sup

≤ kA k

sup

+ kBk

sup

.

Proof. 1. Let A ∈ [[A → B]]

R

. Suppose that

j

:

= inf J(A ) < 0. Then there exists 0 < ε < |j|

such that j + ε ∈ J(A ), namely

− (|j| − ε)C  A  (|j| − ε)C

⇒ −2(|j| − ε)C  0,

for some C ∈ [[A → B]]

1

, which is absurd. Then

inf J(A ) ≥ 0. Suppose now that J(A ) = 0.

This implies that for every n ∈ N there exists

C

n

∈ [[A → B]]

1

such that

1

n

C

n

 A  −

1

n

C

n

.

Now, by the closure of [[A]]

+

in the operational

norm, and considering that the sequences (

1

n

C

n

±

A ) converge to ±A , it must be ±A ∈ [[A]]

+

.

This is possible if and only if A = 0. 2. The

proof is trivial for µ = 0. Let then µ 6= 0. If

x ∈ J(A ), then xC ± A  0 for some C ∈

[[A → B]]

1

, and thus |µ|(xC ± A )  0, namely

|µ|x ∈ J(µA ). Thus kµA k

sup

≤ |µ|kA k

sup

. For

the same reason, kA k

sup

≤ (1/|µ|)kµA k

sup

, and

ﬁnally kµA k

sup

= |µ|kA k

sup

. 3. Let now x ∈

J(A ) and y ∈ J(B). Then xC ± A  0, and

yD ± B  0, for C , D ∈ [[A → B]]

1

. Thus (x +

y)F ± (A + B)  0, where F

:

= x/(x + y)C +

y/(x+y)D ∈ [[A → B]]

1

. Thus, x+y ∈ J(A +B),

and then kA + Bk

sup

≤ kA k

sup

+ kBk

sup

.

The following property makes ([[A →

A]]

R

, k·k

sup

) a Banach algebra.

Proposition 2. For A ∈ [[B → C]]

R

and B ∈

[[A → B]]

R

, kA Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. Let x ∈ J(A ), and y ∈ J(B). Then there

are C , D such that xC ± A  0, yD ± B  0.

Now, we have

1

2

[(xC + A )(yD − B) + (xC − A )(yD + B)

= xyC D − A B  0

1

2

[(xC + A )(yD + B) + (xC − A )(yD − B)

=xyC D + A B  0,

and then xy ∈ J(A B). Thus, kA Bk

sup

≤

kA k

sup

kBk

sup

.

Lemma 4. Let A ∈ [[A → B]]

R

, and for an arbi-

trary C, let D ∈ [[C → D]]

1

. Then kA ⊗ Dk

sup

=

kA k

sup

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 11

Proof. Let x ∈ J(A ). Then by deﬁnition there

exists C ∈ [[A → B]]

1

such that xC ± A  0.

This implies that

xC ⊗ D ± A ⊗ D  0,

namely x ∈ J(A ⊗ D), and then J(A ) ⊆ J(A ⊗

D). Now, let y ∈ J(A ⊗ D). Then there exists

C

0

∈ [[AC → BD]]

1

such that yC

0

± A ⊗ D  0.

Composing the l.h.s. of the latter relation with

ψ ∈ [[C]]

1

and e

D

, we obtain yC

00

± A  0,

where C

00

:

= (e

D

|C

0

|ψ) is a channel, and then

y ∈ J(A ). Thus, J(A ⊗ D) ⊆ J(A ). Finally,

since J(A ⊗ D) = J(A ), we have kA ⊗ Dk

sup

=

kA k

sup

.

Corollary 2. For A ∈ [[A → B]]

R

, and for arbi-

trary C, it is kA ⊗ I

C

k

sup

= kA k

sup

.

Corollary 3. For A ∈ [[A → B]]

R

and B ∈

[[C → D]]

R

, kA ⊗ Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. The result follows straightforwardly from

proposition 2 and corollary 2.

An important result regarding the sup-norm is

provided by the following proposition.

Proposition 3. For A ∈ [[B → C]]

R

and B ∈

[[A → B]]

R

, kA Bk

op

≤ kA k

sup

kBk

op

.

Proof. By deﬁnition we have

kA Bk

op

= sup

D,Ψ∈[[AD]]

1

sup

a∈[[

¯

C

¯

D]]

(2a − e

CD

|A B ⊗ I

D

|Ψ).

Now, for every a ∈ [[

¯

C

¯

D]], and λ ∈ J(A ), upon

deﬁning a

0

:

= a, a

1

:

= e

CD

− a, we have

(a

0

− a

1

|A ⊗ I

C

= (a

0

|[A ⊗ I

D

] − (a

1

|[A ⊗ I

D

]

= λ[(˜a

0

| − (˜a

1

|],

where (˜a

i

|

:

= λ

−1

{(a

i

|[A ⊗I

D

]+

1

2

(e|[(λC −A )⊗

I

D

]}, with C : [[B → C]]

1

such that λC ± A  0.

Clearly,

˜a

0

, ˜a

1

∈ [[

¯

B

¯

D]], ˜a

0

+ ˜a

1

= e

BD

,

thus

(a

0

− a

1

|A B ⊗ I

D

|Ψ)

= λ[(˜a

0

| − (˜a

1

|]B ⊗ I

D

|Ψ)

≤ λ sup

b∈[[

¯

B

¯

D]]

(2b − e

BD

|B ⊗ I

D

|Ψ)

This implies that kA Bk

op

≤ λ sup

b∈[[

¯

B

¯

D]]

(2b −

e

BD

|B ⊗ I

D

|Ψ), and then for every λ ∈ J(A )

kA Bk

op

≤ λ sup

D,Ψ∈[[AD]]

1

sup

b∈[[

¯

B

¯

D]]

(2b − e

BD

|B ⊗ I

D

|Ψ)

= λkBk

op

.

Finally, taking the inﬁmum over λ ∈ J(A ) we

get

kA Bk

op

≤ kA k

sup

kBk

op

.

Corollary 4. The sup-norm is stronger than the

operational norm.

Proof. It is suﬃcient to observe that kA k

op

=

kA I k

op

≤ kA k

sup

kI k

op

= kA k

sup

.

Lemma 5. Let A ∈ [[A → B]]

1

be a channel.

Then kA k

sup

= 1.

Proof. Clearly, A  A  −A , thus 1 ∈ J(A ),

and inf J(A ) ≤ 1. On the other hand, suppose

that there exists 1 > λ ∈ J(A ). This implies

that there exists a channel C ∈ [[A → B]]

1

such

that D

:

= λC − A  0. However, this implies

that (e|

B

D = −(1 − λ)(e|

A

 0. This is ab-

surd, and then λ ∈ J(A ) must be λ ≥ 1. Thus,

inf J(A ) ≥ 1. Finally, this implies kA k

sup

=

1.

Corollary 5. The sup-norm of the identity chan-

nel I ∈ [[A → A]]

1

is 1.

The sup-norm for eﬀects is just the special case

of the sup-norm of transformations with the out-

put system equal to I.

Deﬁnition 7. Let a ∈ [[

¯

A]]

R

, and let us deﬁne

the half-line

J(a)

:

= {λ ∈ R

+

| λe

A

 a  −λe

A

}.

The sup-norm kak

sup

is deﬁned as

kak

sup

:

= inf J(a).

Proposition 4. The sup-norm on [[

¯

A]]

R

is well

deﬁned:

1. kak

sup

is non-negative, and it is null iﬀ a =

0,

2. for µ ∈ R one has kµak

sup

= |µ|kak

sup

,

3. ka + bk

sup

≤ kak

sup

+ kbk

sup

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 12

Proof. The case of eﬀects is just a special case of

the result of Proposition 1.

As an immediate consequence of proposition 3,

we have the following result.

Corollary 6. The sup-norm is stronger than the

operational norm on [[

¯

A]]

R

.

In the special case of [[I → I]]

R

= R, one has

[[I → I]]

1

= {1}, and [[I → I]]

+

= R

+

, thus J(x) =

{λ ∈ R | λ±x ≥ 0}. Clearly, kxk

sup

= inf J(x) =

|x|. We now need the following lemma.

Lemma 6. Let a ∈ [[

¯

A]]

R

. Then ka ⊗ e

B

k

op

=

kak

op

and ka ⊗ e

B

k

sup

= kak

sup

.

Proof. For k·k

op

, the equality trivially follows

from the deﬁnition. For k·k

sup

, it is a special

case of lemma 4.

From lemma 5, the following consequence fol-

lows.

Corollary 7. For the deterministic eﬀect e

A

one

has ke

A

k

sup

= ke

A

k

op

= 1.

We now prove a result that will be very useful

later.

Lemma 7. Let A ∈ [[A → B]]

+

, and (a|

:

=

(e

B

|A ∈ [[

¯

A]]

+

. Then kA k

sup

= kak

sup

.

Proof. First of all, by proposition 2, one has

kak

sup

≤ ke

A

k

sup

kA k

sup

= kA k

sup

. On the

other hand, let λ ∈ J(a). This implies that

b

:

= λe

A

− a  0.

Let us then deﬁne B

:

= |ρ)(b| ∈ [[A → B]]

+

, for

some ρ ∈ [[B]]

1

. By theorem 3, it is easy to check

that A + B = λC for C ∈ [[A → B]]

1

. Then,

λC − A = B  0, (9)

which means that λ ∈ J(A ). Then, J(a) ⊆

J(A ), and ﬁnally kA k

sup

≤ kak

sup

.

In the case of Classical and Quantum Theory,

phrasing the deﬁnitions of sup- and operational

norm in terms of semideﬁnite programming prob-

lems, one can conclude that they are strongly

dual. As a consequence, in these special cases

they deﬁne the same norm.

4 The quasi-local algebra in OPTs

Following the deﬁnition of Ref. [31], we de-

ﬁne a cellular automaton in a general OPT as

a triple (G, A, V ), where G is a denumerable

set of labels for the systems that compose the

automaton—addresses of the memory cells—, A

G

is a (possibly inﬁnite) composite system corre-

sponding to the collection of systems A

g

labelled

by elements g ∈ G—i.e. the memory array—, and

V is a reversible transformation on [[

¯

A

G

]], such

that V · V

−1

is an automorphism of [[A

G

→ A

G

]].

However, this is deﬁnition is incomplete, for many

reasons. In the ﬁrst place, most of the objects

mentioned above are not thoroughly deﬁned. The

purpose of the present section is to set the ground

for the rigorous deﬁnition of a cellular automaton.

We start ﬁxing some notation. For every R ⊆

G let

A

R

:

=

O

g∈R

A

g

,

with the convention that A

∅

:

= I. The full sys-

tem is then A

G

=

N

g∈G

A

g

. This purely formal

notion will now be thoroughly substantiated.

With a slight abuse of notation, we will often

use R = g instead of R = {g}, dropping the

braces. The set of ﬁnite regions of G will be de-

noted as

R

(G)

:

= {R ⊆ G | |R| < ∞},

while the set of arbitrary regions of G will be

denoted as

R

(G)

:

= {R ⊆ G}.

Clearly R

(G)

⊆ R

(G)

. In the remainder, when we

write e.g. e

R

or I

R

for R ∈ R

(G)

, we mean the

deterministic eﬀect or the identity transformation

for the system A

R

, respectively.

The following construction of

N

g∈G

A

g

is in-

spired by that of Refs. [49, 50], however with sig-

niﬁcant diﬀerences.

4.1 Quasi-local eﬀects

In this subsection we start the mathematical con-

struction of the system A

G

, the parallel compo-

sition of inﬁnitely many ﬁnite systems. The ﬁrst

object that we will deﬁne is the space of gener-

alised eﬀects, along with its convex cone of pos-

itive eﬀects, and the convex set of eﬀects. The

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 13

construction is based on the notion of a local ef-

fect, that is an eﬀect which acts non trivially only

on ﬁnitely many systems labelled by g ∈ R, with

R ∈ R

(G)

. We will say that such an eﬀect acts

on the ﬁnite region R. We will then introduce a

real vector space structure over the set of local

eﬀects, and a norm that is induced by the sup

norm for local eﬀects. The last step is then the

topological closure of the space of local eﬀects in

the sup-norm. The Banach space thus obtained

is the space of generalised quasi-local eﬀects, and

the positive cone along with the convex set of ef-

fects contain the limits of sequences of elements of

local cones or convex sets of eﬀects, respectively.

As we will see in the next section, the deﬁnition

of the space of quasi-local states is much more in-

volved than that of quasi-local eﬀects. The latter

is particularly simple, thanks to causality, that

provides us with a preferential local eﬀect, the

unique deterministic one, without the need of in-

troducing any arbitrary choice of a reference local

eﬀect. Moreover, while the construction of the

space of quasi-local states is not logically nec-

essary, as we will ﬁnd them as a subspace of

bounded functionals on quasi-local eﬀects, the

same is not true of eﬀects.

If we deﬁned eﬀects as the dual space of quasi-

local states, we would end up with far more eﬀects

than needed, while lacking sectors of the state

space, that are usually reached by the evolution of

a quasi-local state through a cellular automaton.

These are the main reasons why our construction

begins with eﬀects.

The ﬁrst notion we introduce is that of a lo-

cal eﬀect. Let A

G

denote the formal composition

of countably many systems from a general OPT:

A

G

:

=

N

g∈G

A

g

. Intuitively speaking, a local ef-

fect of A

G

is an event within a test that discards

all the systems in G but the ﬁnite region R, where

a non-trivial measurement is performed. We then

deﬁne the local eﬀect (a, R) as a pair made of an

eﬀect a ∈ [[

¯

A

R

]] and the region R. To lighten the

notation, in the remainder of the paper we will

use the symbol a

R

instead of (a, R). The set of

local eﬀects is denoted by

Pre[[

¯

A

G

]]

L

:

=

G

R∈R

(G)

[[

¯

A

R

]] = {a

R

| R ∈ R

(G)

, a ∈ [[

¯

A

R

]]}.

The above deﬁnition can be widened encompass-

ing generalised eﬀects:

Pre[[

¯

A

G

]]

LR

:

=

G

R∈R

(G)

[[

¯

A

R

]]

R

.

Let us now consider a partition of the region R

into two disjoint regions S

0

∪ S

1

= R. Since in

the deﬁnition of a local eﬀect a

R

we did not set

constraints on the eﬀect a ∈ [[

¯

A

R

]]

R

, it might be

that a = e

S

0

⊗ a

0

, with a

0

∈ [[

¯

A

S

1

]]

R

. It is clear

from our intuitive notion of a local eﬀect that a

R

and a

0

S

1

should represent the same eﬀect. This

observation leads us to the equivalence relation

deﬁned as follows.

Deﬁnition 8 (Equivalent local eﬀects). We say

that the eﬀects a

R

and a

0

S

in Pre[[

¯

A

G

]]

LR

are

equivalent, and denote this as a

R

∼ a

0

S

, if there

exists a

0

∈ [[

¯

A

R∩S

]]

R

such that the following iden-

tities hold

(

a = a

0

⊗ e

S\R

,

a

0

= a

0

⊗ e

R\S

.

(10)

It is clear that the real notion of a local ef-

fect is captured by the equivalence classes modulo

the above equivalence relation. We thus quotient

Pre[[

¯

A

G

]]

R

and deﬁne the obtained set as the set

of local eﬀects of A

G

.

Deﬁnition 9. A generalised local eﬀect is an

equivalence class [a

R

]

∼

. The set of generalised

local eﬀects of A

G

is

[[

¯

A

G

]]

LR

:

= Pre[[

¯

A

G

]]

LR

/ ∼ .

With a slight abuse of notation, in the following

we will write a

R

instead of [a

R

]

∼

, unless the con-

text requires explicit distinction of the two sym-

bols. One can easily prove the following result

Lemma 8. Let a

R

∈ Pre[[

¯

A

G

]]

LR

. Then for every

ﬁnite region H ∈ R

(G)

such that H ∩ R = ∅,

one has (a ⊗ e

H

)

R∪H

∈ Pre[[

¯

A

G

]]

LR

, and (a ⊗

e

H

)

R∪H

∼ a

R

.

The proof of the above lemma is straightfor-

ward, and we do not report it here.

Let us now come back to our initial goal, that

is to deﬁne a local eﬀect as an event that acts

non-trivially only on a ﬁnite region R. Intuition

leads again to ﬁgure out what is the preferred

representative of the class of a local eﬀect: within

the equivalence class, it is the element deﬁned on

the smallest region. We now provide a formal

deﬁnition of such a minimal representative, and

show that it is well posed.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 14

Deﬁnition 10. The minimal representative of

the equivalence class a

R

, denoted as ˜a

R

a

, is de-

ﬁned through

R

a

:

=

\

S∈R

(a,R)

S, ˜a

R

a

∼ a

R

, (11)

where R

(a,R)

is the set of all those ﬁnite regions

S ∈ R

(G)

for which there exists b ∈ [[

¯

A

S

]]

R

such

that b

S

∼ a

R

.

Lemma 9. The minimal representative exists

and is unique.

Proof. As to existence, we remark that, by

Eq. (11), R

a

is the set of all g ∈ G such that

for all regions S ∈ R

(a,R)

one has g ∈ S. Thus,

if h 6∈ R

a

, there must exist S ∈ R

(a,R)

such that

h 6∈ S. This implies that i) by deﬁnition there

exists f

S

∼ a

R

, and ii) by lemma 8 c

S∪h

∼ a

R

,

with

c = f ⊗ e

h

. (12)

As a consequence, if c

T

∼ a

R

and h ∈ T , but

h 6∈ R

a

, by deﬁnition (10) c

T

must be of the

form of Eq. (12), for some f ∈ [[

¯

A

T \h

]]

R

. Clearly,

R

a

∈ R

(G)

, since for any S ∈ R

(a,R)

one has R

a

⊆

S. Now, let S ∈ R

(a,R)

, and c

S

∼ a

R

. One has

S = R

a

∪ S

0

, where S

0

:

= (S \ R

a

) ∈ R

(G)

, thus

R

a

∩ S

0

= ∅. By Eq. (12), we then have

c = b ⊗ e

S

0

, b ∈ [[

¯

A

R

a

]]

R

. (13)

We now deﬁne ˜a

R

a

:

= b

R

a

, and ﬁnally, since

Eq. 13 holds for any c

S

∼ a

R

, one can easily

verify that ˜a

R

a

∼ a

R

.

As to uniqueness, we remark that the re-

gion R

a

is uniquely deﬁned. Now, suppose that

there were two diﬀerent b, c ∈ [[

¯

A

R

a

]]

R

such that

b

R

a

, c

R

a

∼ a

R

. Then by Eq. (10) it must be

b = c.

We now make local eﬀects into a vector space.

Deﬁnition 11. Let a

R

, b

S

∈ [[

¯

A

G

]]

LR

, and h ∈ R.

Then we deﬁne

ha

R

:

=

(

(ha)

R

h 6= 0,

0

∅

h = 0,

a

R

+ b

S

:

= c

R∪S

c

:

= a ⊗ e

S\R

+ b ⊗ e

R\S

.

Notice that it is not always true that R

c

= R

a

∪

R

b

. As an example, consider a = f

g

1

⊗f

g

2

and b =

f

g

1

⊗ (e − f

g

2

), with f

g

1

6= e

g

1

, 0 6= f

g

2

6= e

g

2

, and

R

a

= R

b

= {g

1

, g

2

}. Then c = a + b = f

g

1

⊗ e

g

2

,

and clearly R

c

= {g

1

}, which is strictly included

in R

a

= R

b

= R

a

∪ R

b

.

It is easy to check that [[

¯

A

G

]]

LR

is a real vector

space with null element given by the equivalence

class of 0

∅

.

We now equip the real vector space of local

eﬀects with a norm, and we then close it to ob-

tain the Banach space of quasi-local eﬀects. The

deﬁnition is based on a norm for eﬀects of ﬁnite

systems, as every local eﬀect a

R

∈ [[

¯

A

G

]]

LR

re-

duces to the eﬀect ˜a ∈ [[

¯

A

R

a

]]

R

. A natural norm

one might think of is then the operational norm,

that induces the following deﬁnition.

Deﬁnition 12. The operational norm ka

R

k

op

of

a

R

∈ [[

¯

A

G

]]

LR

is deﬁned by the following expres-

sion

ka

R

k

op

:

= k˜ak

op

, (14)

where ˜a ∈ [[

¯

A

R

a

]]

R

.

Unfortunately, if one completes the space of

local eﬀects in the operational norm, in general

the dual norm on the space of bounded linear

functionals—which will be our state space—does

not coincide with the operational norm on the

state space, for those states that can be inter-

preted as quasi-local preparations. We will then

choose a diﬀerent norm on our space of eﬀects.

The new norm will be referred to as sup-norm, as

it is the extension of the sup-norm to the inﬁnite

case.

As far as ﬁnite-dimensional systems are con-

cerned, this choice does not represent a problem,

as all norms are equivalent in ﬁnite-dimensional

vector spaces. However, the sup-norm is stronger

than the operational one—see corollary 4—, and

thus the space that we construct, completing our

normed vector space of local eﬀects with sup-

norm Cauchy sequences, might contain distinct

limits that are operationally equivalent. This

point is a delicate one, and to avoid an unreason-

able construction where there exist diﬀerent ef-

fects that are operationally equivalent, we impose

a suﬃcient constraint for the operational norm

and the sup-norm to be equivalent also for inﬁ-

nite systems: we restrict attention to those the-

ories where the following property holds: there

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 15

exists a ﬁnite constant k such that for every sys-

tem A and every a ∈ [[

¯

A]]

R

kak

sup

≤ kkak

op

. (15)

This implies that not only the bound (15) holds

for a ﬁxed system A, but it holds with a ﬁxed

constant independent of the system A. In turn,

a suﬃcient condition for (15) is that for every

system A

[[A]]

∗

+

≡ [[

¯

A]]

+

, (16)

namely every positive functional on the cone of

states is proportional to an eﬀect by a positive

constant, and viceversa. In all the presently

known theories the latter condition is satisﬁed.

The two above conditions in Eqs. (15) and (16)

are discussed in detail in Appendix A, where we

prove that condition (16) implies condition (15).

Our new norm is an order-unit norm. As we

will discuss in subsection 4.2, its dual coincides

with the operational norm on quasi-local states.

Let us now see the deﬁnition of the sup-norm

in detail.

Deﬁnition 13. The sup-norm ka

R

k

sup

of a

R

∈

[[

¯

A

G

]]

LR

is deﬁned by the following expression

ka

R

k

sup

:

= k˜ak

sup

, (17)

where ˜a ∈ [[

¯

A

R

a

]]

R

.

We now want to prove that the operational

norm and the sup-norm are well-deﬁned norms

on [[

¯

A

G

]]

LR

.

Proposition 5. Let a

R

∼ ˜a

R

a

. Then ka

R

k

op

=

kak

op

, and ka

R

k

sup

= kak

sup

.

Proof. Let a

R

∼ ˜a

R

a

, and R = R

a

∪ R

0

, with

R

a

∩ R

0

= ∅. Then by Eq. (10) we have

a = ˜a ⊗ e

R

0

,

and thus by deﬁnitions 12 and 13 and lemma 6,

it is

ka

R

k

op

= k˜ak

op

= k˜a ⊗ e

R

0

k

op

= kak

op

,

ka

R

k

sup

= k˜ak

sup

= k˜a ⊗ e

R

0

k

sup

= kak

sup

.

We can then prove the desired result.

Proposition 6. The functionals k·k

op

and k·k

sup

are norms on [[

¯

A

G

]]

LR

.

Proof. 1. For every a

R

∈ [[

¯

A

G

]]

LR

it is clear that

ka

R

k

op

≥ 0 and ka

R

k

sup

≥ 0. Now, k˜ak

op

=

k˜ak

sup

= 0 if and only if ˜a = 0, namely, re-

minding deﬁnition 11, a

R

= 0

∅

. 2. For every

a

R

∈ [[

¯

A

G

]]

LR

, by deﬁnitions 7, 11, and 12, one

straightforwardly has that, for every µ ∈ R,

kµa

R

k

op

= kµ˜ak

op

= |µ|k˜ak

op

= |µ|ka

R

k

op

,

kµa

R

k

sup

= kµ˜ak

sup

= |µ|k˜ak

sup

= |µ|ka

R

k

sup

.

3. Let now c

R∪S

= a ⊗ e

S\R

+ e

R\S

⊗ b for

a ∈ [[

¯

A

R

]]

R

and b ∈ [[

¯

A

S

]]

R

, as from deﬁnition 11.

Then, by lemma 6, proposition 5 and by the tri-

angle inequality, we have

ka

R

+ b

S

k

op

= kc

R∪S

k

op

= kck

op

≤ kak

op

+ kbk

op

= ka

R

k

op

+ kb

S

k

op

,

ka

R

+ b

S

k

sup

= kc

R∪S

k

sup

= kck

sup

≤ kak

sup

+ kbk

sup

= ka

R

k

sup

+ kb

S

k

sup

.

We remark that, in a theory that satisﬁes as-

sumption (16), one has k·k

op

≡ k·k

sup

on [[

¯

A]]

R

,

and thus k·k

op

= k·k

sup

on [[

¯

A

G

]]

R

(the proof can

be found in appendix A). In general, however, the

sup-norm is stronger than the operational norm,

as we now prove.

Lemma 10. Let a

R

∈ [[

¯

A

G

]]

LR

. Then ka

R

k

op

≤

ka

R

k

sup

.

Proof. Let us consider µ ∈ J(a). Then for every

ρ ∈ [[A

R

]]

1

we have

(µe

R

± a|ρ) ≥ 0,

i.e. |(a|ρ)| ≤ µ. Then, taking the supremum over

states on l.h.s. we obtain ka

R

k

op

≤ µ, and ﬁnally,

taking the inﬁmum over J(a) we obtain the the-

sis.

The space of local eﬀects that we constructed

so far is a normed real vector space. We now

make it into a Banach space, the space of quasi-

local eﬀects, by the usual completion procedure

for normed spaces. We ﬁrst introduce the space

[[

¯

A

G

]]

CR

of Cauchy sequences

a : N → [[

¯

A

G

]]

LR

:: n 7→ a

n

R

n

.

We then deﬁne the equivalence relation between

Cauchy sequences in [[

¯

A

G

]]

CR

deﬁned by a

∼

=

b

iﬀ lim

n→∞

ka

n

R

n

− b

n

S

n

k

sup

= 0. Finally, we de-

ﬁne the space [[

¯

A

G

]]

QR

of generalised quasi-local

eﬀects, by taking the quotient

[[

¯

A

G

]]

QR

:

= [[

¯

A

G

]]

CR

/

∼

=

. (18)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 16

The elements of this space will be denoted by

a = [a

n

R

n

]. In this context, the class of a con-

stant Cauchy sequence (R

n

= R and a

n

R

n

= a

R

)

will be denoted by a

R

:

= [a

n

R

n

]. Clearly, since

|ka

n

R

n

k

sup

− ka

m

R

m

k

sup

| ≤ ka

n

R

n

− a

m

R

m

k

sup

,

for a Cauchy sequence a = [a

n

R

n

] also ka

n

R

n

k

sup

is a Cauchy sequence, and we deﬁne kak

sup

:

=

lim

n→∞

ka

n

R

n

k

sup

. One can easily verify that

the sup-norm on [[

¯

A

G

]]

QR

thus deﬁned is indepen-

dent of the speciﬁc sequence within an equiva-

lence class. The space [[

¯

A

G

]]

QR

is by construction

a real Banach space, and [[

¯

A

G

]]

LR

can be identi-

ﬁed with the dense submanifold containing con-

stant sequences a

R

= [a

R

]. Clearly, in this case

kak

sup

= ka

R

k

sup

.

Deﬁnition 14. Let a ∈ [[

¯

A

G

]]

QR

. If there is a

Cauchy sequence a

n

R

n

in the class deﬁning a,

such that, for some n

0

∈ N, for every n ≥ n

0

one has ˜a

n

∈ [[

¯

A

R

n

]], then we call a a quasi-local

eﬀect. We denote the set of quasi-local eﬀects by

[[

¯

A

G

]]

Q

.

The cone [[

¯

A

G

]]

Q+

contains all elements that

are proportional to an element in [[

¯

A

G

]]

Q

by a pos-

itive constant.

The deterministic quasi-local eﬀect e

G

is the

class e

G

:

= 1

∅

.

The cone [[

¯

A

G

]]

Q+

introduces a partial ordering

in [[

¯

A

G

]]

QR

, that we denote by

a  b ⇔ a − b ∈ [[

¯

A

G

]]

Q+

.

Quasi-local eﬀects can be interpreted as eﬀects

that can be arbitrarily well approximated by local

observation procedures. The eﬀect e

G

plays an

important role, as it is the unique deterministic

eﬀect in [[

¯

A

G

]]

Q

. This will be proved shortly. Let

us start providing the ﬁrst important property of

e

G

.

Lemma 11. Let a ∈ [[

¯

A

G

]]

Q

. Then also e

G

− a ∈

[[

¯

A

G

]]

Q

.

Proof. Let a = [a

n

R

n

]. Let a

0

n

R

n

:

= (e − a

n

)

R

n

∈

[[

¯

A

R

n

]]. Then one has (a

0

n

− a

0

m

)

R

n

∪R

m

= (a

m

−

a

n

)

R

m

∪R

n

, and thus a

0

n

R

n

is a Cauchy sequence

whose class we call a

0

∈ [[

¯

A

G

]]

Q

. Finally, since

a+a

0

is the class [a

n

R

n

]+[a

0

n

R

n

] = [(a

n

+a

0

n

)

R

n

] =

[1

∅

] = e

G

, we have that a + a

0

= e

G

, namely

e

G

− a = a

0

∈ [[

¯

A

G

]]

Q

.

By the above lemma we know that not only

e

G

 a for every quasi-local eﬀect a, but also that

every quasi-local eﬀect a can be complemented

by a quasi-local eﬀect a

0

such that a + a

0

= e

G

.

In other words (a, e

G

− a) is a binary quasi-local

observation test.

We can also prove the converse result.

Lemma 12. Let a ∈ [[

¯

A

G

]]

Q+

, and e

G

− a ∈

[[

¯

A

G

]]

Q+

. Then a, e

G

− a ∈ [[

¯

A

G

]]

Q

.

Proof. Indeed, the statement is true for ﬁnite

systems thanks to the no-restriction hypothesis

(see theorem 5), and thus it holds for a given

Cauchy sequence in the class of a, namely if

a

0

n

R

n

:

= (e − a

n

)

R

n

∈ [[

¯

A

R

n

]]

+

for all n, then

a

n

, a

0

n

∈ [[

¯

A

R

n

]]. Thus, a = lim

n→∞

a

n

∈ [[

¯

A

G

]]

Q

,

and a

0

= lim

n→∞

a

0

n

= e

G

− a ∈ [[

¯

A

G

]]

Q

.

We now prove that e

G

lies in the interior of

[[

¯

A

G

]]

Q+

, namely there is a ball of radius r around

e

G

in sup-norm that is contained in [[

¯

A

G

]]

Q+

.

Lemma 13. There exists an open ball B

r

(e

G

)

of radius r in [[

¯

A

G

]]

QR

that is fully contained in

[[

¯

A

G

]]

Q+

.

Proof. Let ka − e

G

k

sup

< r < 1/2. Then, by

deﬁnition, there exist a sequence a

n

R

n

such that

lim

n→∞

a

n

= a, and n

0

∈ N such that for n ≥ n

0

ka

n

− e

G

k

sup

≤ ka

n

− ak

sup

+ ka − e

G

k

sup

< 2r.

This implies that

[2re + (a

n

− e)]

R

n

= [a

n

− (1 − 2r)e]

R

n

 0.

Now, this implies that for n ≥ n

0

it is a

n



(1 − 2r)e

R

n

 0, and thus a

n

∈ [[

¯

A

G

]]

L+

. Finally,

by deﬁnition, one obtains a ∈ [[

¯

A

G

]]

Q+

. Thus, the

ball B

r

(e

G

) is contained in [[

¯

A

G

]]

Q+

.

Corollary 8. Let a ∈ [[

¯

A

G

]]

Q+

. Then for ε > 0,

B

εr

(εe

G

+ a) ⊆ [[

¯

A

G

]]

Q+

for some r > 0.

Proof. Let b ∈ B

εr

(εe

G

+ a). Then

kb − a − εe

G

k

sup

< εr

⇔ k(b − a)/ε − e

G

k

sup

< r.

Then (b − a)/ε = c ∈ [[

¯

A

G

]]

Q+

, ad thus

b = εc + a ∈ [[

¯

A

G

]]

Q+

.

We now prove that the deterministic eﬀect e

G

allows for an extension of the deﬁning property

of the sup-norm.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 17

Lemma 14. The sup-norm of a ∈ [[

¯

A

G

]]

QR

can

be expressed as

kak

sup

= inf J(a),

J(a)

:

= {µ ≥ 0 | µe

G

 a  −µe

G

}.

Proof. In the case of local eﬀects a = a

R

the

statement is a trivial recasting of the deﬁnition

of sup-norm. Let us then consider the class a ∈

[[

¯

A

G

]]

QR

with a = [a

n

R

n

]. One has, by deﬁnition,

kak

sup

= lim

n→∞

ka

n

k

sup

. Let us consider the

sequence µ

n

:

= ka

n

k

sup

+ ε ∈ J(a

n

). Clearly,

lim

n→∞

µ

n

= kak

sup

+ ε. Moreover, since

k{(µ

n

− µ

m

)e ± (a

n

− a

m

)}

R

n

∪R

m

k

sup

≤ |ka

m

k

sup

− ka

n

k

sup

| + ka

m

− a

n

k

sup

≤ 2ε,

the sequence {(µ

n

+ ε)e ± a

n

}

R

n

converges to

(kak

sup

+ε)e

G

±a ∈ [[

¯

A

G

]]

Q+

, thus for every ε ≥ 0

one has kak

sup

+ ε ∈ J(a). This implies that

inf J(a) ≤ kak

sup

.

On the other hand, let µ ∈ J(a), then

µe

G

± a  0.

By corollary 8, we have that, for ε > 0,

(ε + µ)e

G

± a ∈ B

εr

(εe

G

+ µe

G

± a) ⊆ [[

¯

A

G

]]

Q+

.

Now, this implies that there are two open balls,

centred at (µ + ε)e

G

± a, that are fully contained

in [[

¯

A

G

]]

Q+

. There exists then n

0

∈ N such that

for n ≥ n

0

one has

(µ + ε)e

G

± a

n

 0,

namely (µ + ε) ∈ J(a

n

). Thus, J(a) ⊆ J(a

n

) − ε,

and

inf J(a) ≥ ka

n

k

sup

− ε. (19)

Finally, this implies that inf J(a) ≥ kak

sup

.

As the cone [[

¯

A

G

]]

Q+

is closed, we can easily

prove that the inﬁmum in the expression of the

sup norm is actually a minimum.

Lemma 15. Let a ∈ [[

¯

A

G

]]

QR

. Then

kak

sup

e

G

± a  0. (20)

Proof. By lemma 14, for every ε > 0 one has

(kak

sup

+ ε)e

G

± a  0.

The sequences b

±

n

:

= (kak

sup

+1/n)e

G

±a are both

Cauchy, and their limits are both in [[

¯

A

G

]]

Q+

,

then

kak

sup

e

G

± a  0.

Before concluding, we remark that the opera-

tional norm can be extended to [[

¯

A

G

]]

QR

, by sim-

ply deﬁning for a = [a

n

R

n

]

kak

op

:

= lim

n→∞

ka

n

R

n

k

op

.

Indeed, since

|ka

n

R

n

k

op

− ka

m

R

m

k

op

| ≤ ka

n

R

n

− a

m

R

m

k

op

≤ ka

n

R

n

− a

m

R

m

k

sup

,

the sequence ka

n

R

n

k

op

is Cauchy, and the limit

is well deﬁned. Moreover, also on [[

¯

A

G

]]

QR

the sup-norm is stronger than the operational

norm, as can be straightforwardly concluded from

lemma 10. However, in theories where there ex-

ists k > 0 such that k·k

sup

≤ kk·k

op

in [[

¯

A]]

R

for

every ﬁnite system A, the same inequality holds

in the limit, and the operational and sup-norm

are equivalent. In particular, this is the case un-

der assumption (16).

We now introduce a diagrammatic notation for

eﬀects in [[

¯

A]]

QR

that will make part of the subse-

quent proofs and arguments more intuitive. First

of all, we denote a quasi-local eﬀect a ∈ [[

¯

A

G

]]

QR

by the symbol

G

a

. (21)

In the case of a local eﬀect a

R

∈ [[

¯

A

G

]]

LR

, we will

draw

G

a

R

=

R

a

G\R

e

. (22)

Notice that, since for (a ⊗ e

R

1

)

R

with a ∈ [[

¯

A

R

0

]]

and R

1

:

= R \ R

0

, it is (a ⊗ e

R

1

)

R

∼ a

R

0

, we have

R

0

a

R

1

e

G\R

e

=

R

0

a

(G\R)∪R

1

e

. (23)

From the above identity, we can intuitively con-

clude that

(G\R)∪R

1

e

=

R

1

e

G\R

e

. (24)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 18

Indeed, let G = R ∪ H, with |R| < ∞. Then

[e

R

] = [1

∅

] = e

G

, which is the equation repre-

sented by the diagram in Eq. 24. Similarly, let

a

R

⊗ b

H

:

= lim

n→∞

(a ⊗ b

n

), where [b

nS

n

] = b ∈

[[

¯

A

H

]]

QR

. By corollary 3, one has [(a⊗b

n

)

R∪S

n

] =

(a

R

⊗ b

H

). Thus, in G = R ∪ H, one has

a

R

= [a

R

] = [(a

R

⊗ e

H

)] = a

R

⊗ e

H

, which is

the meaning of Eq. (22).

As a ﬁnal remark, we observe that for every

denumerable set G, and every subset R ⊆ G, in-

cluding inﬁnite ones, one can deﬁne [[

¯

A

(G)

R

]]

QR

as

the closed subspace spanned by those quasi-local

eﬀects a = a

n

R

n

such that, for every n ∈ N, R

n

⊆

R. If one constructs the system A

R

:

=

N

g∈R

A

g

,

it is straightforward to construct an ordered Ba-

nach space isomorphism

J

†

R

: [[

¯

A

(G)

R

]]

QR

→ [[

¯

A

R

]]

QR

:: [a

n

R

n

]

G

7→ [a

n

R

n

]

R

,

(25)

with the norm coinciding with the sup-norm in

both spaces. The left-inverse of J

†

R

is

J

−1†

R

: [[

¯

A

R

]]

QR

→ [[

¯

A

(G)

R

]]

QR

. (26)

J

†

R

and J

−1†

R

can be diagrammatically denoted

as

R

J

†

R

a

G\R

e

=

R

a

,

(27)

G

J

−1†

R

a

=

R

a

G\R

e

. (28)

4.2 Extended states

Now that we deﬁned the space of quasi-local ef-

fects, we can deﬁne the space of states as the

space of bounded linear functionals on [[

¯

A

G

]]

QR

.

The set of states is then deﬁned considering those

linear functionals that, acting on local eﬀects of

an arbitrary ﬁnite region R, behave as a state in

[[A

R

]].

Deﬁnition 15 (Generalised extended states).

The space [[A

G

]]

R

of generalised extended states

of A

G

is the topological dual of [[

¯

A

G

]]

QR

, i.e. the

Banach space [[

¯

A

G

]]

∗

QR

of bounded linear function-

als on [[

¯

A

G

]]

QR

, equipped with the norm

kρk

∗

:

= sup

kak

sup

=1

|(a|ρ)|. (29)

The operational norm on [[A

G

]]

R

, deﬁned as

kρk

op

:

= sup

a∈[[

¯

A

G

]]

Q

(2a − e

G

|ρ), (30)

coincides with the norm k·k

∗

, as we now prove.

Lemma 16. Let ρ ∈ [[A

G

]]

R

. Then

kρk

∗

= sup

a∈[[

¯

A

G

]]

Q

(2a − e

G

|ρ). (31)

Proof. Let 0 ≤ ε ≤ kρk

op

. We then have

0 ≤ kρk

op

− ε ≤ (2a − e

G

|ρ),

for some a ∈ [[

¯

A

G

]]

Q

. Invoking lemma 11, one has

e

G

+ (2a − e

G

) = 2a  0,

e

G

− (2a − e

G

) = 2(e

G

− a)  0,

and, by lemma 14, k2a − e

G

k

sup

≤ 1. Then it is

kρk

op

− ε ≤

(2a − e

G

|ρ)

k2a − e

G

k

sup

≤ kρk

∗

.

On the other hand, let us now pick a ∈ [[

¯

A

G

]]

R

with kak

sup

= 1. Then, by lemma 15, e

G

±a  0.

If we deﬁne a

±

:

=

1

2

(e

G

± a), we have a

±

 0,

and

a

+

+ a

−

= e

G

, (a

+

− a

−

) = a, (32)

which by lemma 12 implies that a

±

∈ [[

¯

A

G

]]

Q

.

Then

±(a|ρ) = (2a

±

− e

G

|ρ).

Thus, |(a|ρ)| ≤ kρk

op

, for every ε > 0, and taking

the supremum on the l.h.s. we obtain kρk

∗

≤

kρk

op

.

Technically speaking, the two norms k·k

sup

and

k·k

∗

are a base and order-unit norm pair [51].

What is missing now is the notion of a convex

set of proper states, the extended preparations of

our inﬁnite system A

G

. Let us then give the fol-

lowing deﬁnition.

Deﬁnition 16 (Local restriction). Given a state

ρ ∈ [[A

G

]]

R

, the local restriction of ρ to S ∈ R

(G)

is the functional ρ

|S

∈ [[A

S

]]

R

deﬁned through

(a|ρ

|S

)

:

= (a

S

|ρ), ∀a ∈ [[

¯

A

S

]]. (33)

The notion of a restriction can be brought fur-

ther, considering inﬁnite regions, by invoking the

isomorphism J

R

deﬁned in Eq. (25), as follows.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 19

Deﬁnition 17 (Restriction). Given a state ρ ∈

[[A

G

]]

R

, the restriction of ρ to S ∈ R

(G)

is the

functional ρ

|S

∈ [[A

S

]]

R

deﬁned through

(a|ρ

|S

)

:

= (J

−1†

S

a|ρ), ∀a ∈ [[

¯

A

S

]]. (34)

Deﬁning

ˆ

J

−1

S

by duality as

(a|

ˆ

J

−1

S

ρ)

:

= (J

−1†

S

a|ρ), (35)

we can then equivalently express Eq. (34) as

ρ

|S

=

ˆ

J

−1

S

ρ. (36)

Representing generalised extended states through

diagrams, and reminding eq. (28), the above

equations can be recast as

ρ

|S

S

a

=

ρ

R

J

−1

S

a

=

ρ

S

a

G\S

e

,

which can be taken as the deﬁnition of the equa-

tion

ρ

|S

S

:

=

ρ

S

G\S

e

. (37)

We can now introduce the set of states [[A

G

]]

as the special set of generalised extended states

whose restrictions are all states. In other words,

an element ρ of the space [[A

G

]]

R

is a state if,

restricted to any ﬁnite region R ∈ R

(G)

, it deﬁnes

a state for that region.

Deﬁnition 18 (Extended states). The set [[A

G

]]

of states in the space of generalised extended

states [[A

G

]]

R

is the set of those elements ρ ∈

[[A

G

]]

R

such that for every R ∈ R

(G)

the local re-

striction of ρ to R is a state ρ

|R

∈ [[A

R

]]. Deter-

ministic states, whose set is denoted by [[A

G

]]

1

,

are those states ρ ∈ [[A

G

]] with (e

G

|ρ) = 1.

One can easily verify that the set [[A

G

]] is con-

vex, as a consequence of convexity of [[A]] for ev-

ery ﬁnite system A. As we did for ﬁnite systems,

we also deﬁne a positive cone generated by [[A

G

]].

Moreover, it is easy to check that the restriciton

of a state to an inﬁnite region S ∈ R

(G)

is s state

in [[A

S

]].

Deﬁnition 19 (Extended positive cone). The set

[[A

G

]]

+

in the space of generalised extended states

[[A

G

]]

R

is the cone of those elements ρ ∈ [[A

G

]]

R

such that there exist ¯ρ ∈ [[A

G

]] and µ ≥ 0 such

that ρ = µ¯ρ.

We can now show an important result about

the restriction of states.

Lemma 17. The restriction

ˆ

J

−1

R

[[A

G

]] of the set

of states of A

G

coincides with [[A

R

]].

Proof. It is a straightforward exercise to verify

that

ˆ

J

−1

R

[[A

G

]] ⊆ [[A

R

]]. For the converse, let

ρ ∈ [[A

R

]], and let us now construct a state σ ∈

[[A

G

]] such that σ

|R

= ρ. We deﬁne

¯

R

:

= G \ R.

There are two possible situations:

¯

R ∈ R

(G)

, or

¯

R ∈ R

(G)

. If

¯

R ∈ R

(G)

, let a

T

∈ [[

¯

A

G

]]

LR

, with

T ∈ R

(G)

. We deﬁne

(a|σ)

:

= (a|ρ

|T ∩R

⊗ ν

|T ∩

¯

R

), (38)

for an arbitrary ν ∈ [[A

¯

R

]], where we introduced

the notation σ

|S

0

for S

0

⊆ S ∈ R

(G)

, and σ a state

of the composite system σ ∈ [[A

S

]] = [[A

S

0

A

S\S

0

]],

with

σ

|S

0

S

0

:

=

σ

S

0

S\S

0

e

.

Clearly, the functional σ is bounded, and thus

it can be extended to a functional on the full

space [[

¯

A

G

]]

QR

, i.e. σ ∈ [[A

G

]]

R

. Moreover one can

easily verify that σ

|T

∈ [[A

T

]] for every T ∈ R

(G)

,

thus σ is the desired state in [[A

G

]]. A similar

construction can be carried out for the case

¯

R ∈

R

(G)

, taking ν ∈ [[A

¯

R

]]. By construction, also in

this case σ ∈ [[A

G

]], as can be straightforwardly

checked.

The next result that we prove is that quasi-

local eﬀects a ∈ [[

¯

A

G

]]

Q

are positive on the set

[[A

G

]].

Lemma 18. Let a ∈ [[

¯

A

G

]]

Q

, and ρ ∈ [[A

G

]].

Then (a|ρ) ≥ 0

Proof. Let a ∈ [[

¯

A

G

]]

Q

be a local eﬀect a

R

=

[a

R

]. Then, by deﬁnition, (a

R

|ρ) = (a|ρ

|R

) ≥

0. Now, let a = [a

n

R

n

]. We have (a|ρ) =

lim

n→∞

(a

n

R

n

|ρ) ≥ 0.

The following result draws a ﬁrst analogy be-

tween the properties of the sup-norm for ﬁnite

systems and that for inﬁnite parallel composi-

tions.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 20

Lemma 19. Let ρ ∈ [[A

G

]]

+

. Then kρk

∗

=

(e

G

|ρ).

Proof. Let ρ ∈ [[A

G

]]

+

. Then, by virtue of

lemma 18, (a|ρ) ≥ 0 for all a ∈ [[

¯

A

G

]]. Let now

kak

sup

= 1. By lemma 15, one has e

G

± a  0,

and thus (e

G

|ρ) ≥ |(a|ρ)|. Then (e

G

|ρ) ≥ kρk

∗

.

On the other hand, being e

G

a legitimate el-

ement of [[

¯

A

G

]]

Q

with ke

G

k

sup

= 1, one has

kρk

∗

= sup

kak

sup

=1

|(a|ρ)| ≥ (e

G

|ρ).

Thanks to lemma 12, we have the following

corollary.

Corollary 9. Let ρ ∈ [[A

G

]]. Then 0 ≤ (a|ρ) ≤

(e

G

|ρ) for every a ∈ [[

¯

A

G

]]

Q

.

We can now show that also in the inﬁnite case

every state is proportional to a deterministic one.

Let us ﬁrst prove a preliminary lemma.

Lemma 20. Let ρ ∈ [[A

G

]]

R

. If (a|ρ) = 0 for

every a ∈ [[

¯

A

G

]]

Q

, then ρ = 0.

Proof. If (a|ρ) = 0 for every a ∈ [[

¯

A

G

]]

Q

then,

using lemma 11, we have

(2a − e

G

|ρ) = (a|ρ) − (e

G

− a|ρ) = 0, ∀a ∈ [[

¯

A

G

]]

Q

.

This implies that kρk

∗

= 0, and thus ρ = 0.

Proposition 7. Every state ρ ∈ [[A

G

]] is propor-

tional to a deterministic state ¯ρ ∈ [[A

G

]]

1

.

Proof. Let ρ ∈ [[A

G

]]. If ρ = 0, then for every

¯ρ ∈ [[A

G

]]

1

it is ρ = 0¯ρ. Let then ρ 6= 0. By

lemmas 18 and 20, there must exist a ∈ [[

¯

A

G

]]

such that (a|ρ) > 0. By corollary 9, one then

has (e

G

|ρ) ≥ (a|ρ) > 0. By deﬁnition, for every

S ∈ R

(G)

, we have

(e

A

S

|ρ

|S

) = (e

S

|ρ) = (e

G

|ρ) > 0.

Thus, if we set ¯ρ

:

= ρ/(e

G

|ρ), for every S ∈ R

(G)

we obtain

(e

A

S

|¯ρ

|S

) = (e

G

|¯ρ) = 1,

implying that ¯ρ

|S

∈ [[A

S

]]

1

for every S ∈ R

(G)

.

Then, ¯ρ ∈ [[A

G

]]

1

, and ¯ρ is such that ρ = (e

G

|ρ)¯ρ.

We now show that the operational norm is

equivalently deﬁned on [[

¯

A

G

]]

QR

by

kak

op

:

= sup

ρ∈[[A

G

]]

|(a|ρ)|.

Indeed, let a = [a

nR

n

]. Then

|(a|ρ)| = lim

n→∞

|(a

nR

n

|ρ)| = lim

n→∞

|(a

n

|ρ

|R

n

)|.

This implies that

|(a|ρ)| ≤ |(a

n

|ρ

|R

n

)| + ε ≤ ka

nR

n

k

op

+ ε,

thus sup

ρ∈[[A

G

]]

|(a|ρ)| ≤ kak

op

. On the other

hand, for every a

nR

n

and every ε, there exists

ρ

ε

∈ [[A

R

n

]] such that ka

nR

n

k

op

≤ |(a

n

|ρ

ε

)| + ε.

Now, by lemma 17, this implies that for every ε

there exists ˜ρ

ε

∈ [[A

G

]] such that

ka

nR

n

k

op

− ε ≤ |(a

nR

n

|˜ρ

ε

)|.

Finally, this implies that for every ε there exists

ρ

ε

∈ [[A

G

]] such that

kak

op

− 3ε ≤ ka

nR

n

k

op

− 2ε

≤ |(a

nR

n

|˜ρ

ε

)| − ε

≤ |(a|˜ρ

ε

)|. (39)

Thus, in conclusion, kak

op

≤ sup

ρ∈[[A

G

]]

|(a|ρ)|.

A crucial property of [[A

G

]] is that extended

states are separating for [[

¯

A

G

]]

Q

, namely if for ev-

ery state two generalised eﬀects give the same

value, then they coincide.

Theorem 6 (States separate eﬀects). Let a ∈

[[

¯

A

G

]]

QR

. If (a|ρ) = 0 for all ρ ∈ [[A

G

]]

1

, then

a = 0.

Proof. Let a = [a

n

R

n

] ∈ [[

¯

A

G

]]

QR

, and (a|ρ) = 0

for every ρ ∈ [[A

G

]]

1

. Then for every ρ ∈ [[A

G

]]

1

one has

|(a

n

|ρ)| = |(a

n

− a|ρ)| ≤ ka

n

− ak

sup

,

and thus for every ε > 0 there exists n

0

such

that, for n ≥ n

0

, ka

n

k

op

≤ ε. Thus, we have

kak

op

= 0. As a consequence, for every ε > 0

there exists n

0

such that for n ≥ n

0

one has

ka

n

k

sup

≤ kka

n

k

op

≤ ε,

and thus kak

sup

= 0. Now, this implies that a =

0.

Remark 2. The above result is made possible

by requirement (15) that the sup-norm and the

operational norm for eﬀects of ﬁnite systems A

are bounded as

∀a ∈ [[

¯

A]]

R

kak

sup

≤ kkak

op

,

with k independent of the system A. In par-

ticular, this is true under the assumption (16),

i.e. that [[

¯

A]]

∗

+

≡ [[A]]

+

.

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We can now prove the main consequence of

uniqueness of the deterministic eﬀect e

G

for A

G

,

i.e. that e

G

is the unique eﬀect that amounts to

1 on [[A

G

]]

1

.

Proposition 8. Let a ∈ [[

¯

A

G

]]

Q

. Then (a|ρ) = 1

for all ρ ∈ [[A

G

]]

1

if and only if a = e

G

.

Proof. Notice that, since e

G

 a for every a ∈

[[

¯

A

G

]]

Q

, and (b|ρ) ≥ 0 for every b ∈ [[

¯

A

G

]]

Q

and

ρ ∈ [[A

G

]]

1

, we have

0 ≤ (e

G

− a|ρ) = 1 − (a|ρ),

which implies (a|ρ) ≤ 1 for every ρ ∈ [[A

G

]]

1

.

Now, if (a|ρ) = 1 for every ρ ∈ [[A

G

]]

1

, we have

(e

G

− a|ρ) = 0, ∀ρ ∈ [[A

G

]]

1

,

which by theorem 6 implies a = e

G

.

A class of states of particular interest is that

of quasi-local states, which are intuitively under-

stood as states whose preparation can be arbi-

trarily approximated by local procedures. These

live in small subspaces of the space [[A

G

]]

R

. Their

construction is similar to that of quasi-local ef-

fects, but diﬀers form it in a relevant respect,

that is the necessity of deﬁning arbitrary refer-

ence states. The construction is inspired by pio-

neering works on inﬁnite tensor products by von

Neumann and Murray [49], and can be found in

appendix B.

4.3 Quasi-local transformations

Now we are going to deﬁne the quasi-local alge-

bra [[A

G

→ A

G

]]

QR

of transformations. In usual

approaches to quantum inﬁnite systems, one usu-

ally introduces the eﬀect algebra, that in the OPT

case would correspond to the space of quasi-local

eﬀects. However, one can easily understand that,

unlike eﬀects in a general OPT, the quantum ef-

fect algebra contains far more information than

the mere structure of the space of eﬀects. Indeed,

in the quantum eﬀect algebra one can ﬁnd every

operator on the Hilbert space, and in turn, the

operational role of a linear operator is to provide a

Kraus representation of a transformation. In this

perspective, one can also understand why mul-

tiplication of eﬀects has a meaning at all: there

is no point in multiplying eﬀects, but multiplica-

tion of Kraus operators is the mathematical rep-

resentation of sequential composition. Thus, the

algebraic structure of eﬀects conveniently sum-

marises information about every kind of event in

the theory: eﬀects, with their coarse-graining rep-

resented by the sum, and transformations, with

sequential composition represented by multiplica-

tion.

Cellular automata are deﬁned in quantum the-

ory by specifying their action on the eﬀect alge-

bra. Such an action thus provides information

about how both eﬀects and transformations are

transformed by the cellular automaton. For a

general OPT, however, such a compact algebraic

structure embodying all the relevant information

is absent, and the deﬁnition of a CA on the space

of eﬀects is not suﬃcient: we need to specify how

the CA transforms transformations.

For these reasons we construct now the Banach

algebra of quasi-local transformations, in a way

that is very closely reminiscent of the construc-

tion of quasi-local eﬀects. Here, in order to deﬁne

a local transformation, we recur to the notion of a

transformation that acts non-trivially on ﬁnitely

many systems, while it acts as the identity on the

remaining ones. In other words, the role that is

played by the deterministic eﬀect in the case of

local eﬀects is played here by the identity trans-

formation. Also in this case, the topological clo-

sure will be taken in the sup-norm, and this choice

is of great relevance to ensure a Banach algebra

structure for the closure.

Let us then introduce the Banach algebra of

quasi-local transformations.

Deﬁnition 20. Let R ⊆ G be an arbitrary ﬁnite

region of G. We deﬁne local transformation A

R

of A

G

any pair (A , R), where A ∈ [[A

R

→ A

R

]]

R

.

The action A

†

R

of A

R

on Pre[[

¯

A

G

]]

LR

is deﬁned as

follows

R\S

A

†

R

a

S

R∩S

S\R

:

=

R\S

A

R

R\S

e

R∩S R∩S

a

S

S\R

. (40)

We also deﬁne Pre[[A

G

→ A

G

]]

LR

as the set of

local transformations A

R

.

Lemma 21. Let a

S

∼ b

T

. Then A

†

R

a

S

∼ A

†

R

b

T

.

Proof. By hypothesis, there exists c ∈ [[

¯

A

S∩T

]]

R

such that

a = c ⊗ e

S\T

, b = c ⊗ e

T \S

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 22

Then we have

(A

†

R

a

S

| = (c ⊗ e

S\T

⊗ e

R\S

|(A ⊗ I

S\R

),

(A

†

R

b

T

| = (c ⊗ e

T \S

⊗ e

R\T

|(A ⊗ I

T \R

).

By Eq. (10) the thesis follows.

We can now deﬁne an equivalence relation be-

tween local transformations as follows

A

S

∼ B

T

⇔

(

A = C ⊗ I

S\T

,

B = C ⊗ I

T \S

,

(41)

for some C ∈ [[A

S∩T

]]

R

. The set of local transfor-

mations is then deﬁned as

[[A

G

→ A

G

]]

LR

:

= Pre[[A

G

→ A

G

]]

LR

/ ∼

Lemma 22. Let A

R

∈ Pre[[A

G

→ A

G

]]

LR

. Then

for every ﬁnite region H ∈ R

(G)

such that H ∩

R = ∅, one has (A ⊗ I

A

H

)

R∪H

∈ Pre[[A

G

→

A

G

]]

LR

, and (A ⊗ I

A

H

)

R∪H

∼ A

R

.

Proof. It is straightforward to verify that (A ⊗

I

A

H

)

R∪H

∼ A

R

by direct inspection of the deﬁn-

ing equation (41).

We can now provide a way to identify a canon-

ical representative of the equivalence class of A

R

,

which is deﬁned in analogy to the case of eﬀects

as follows.

Deﬁnition 21. The minimal representative of

the equivalence class A

R

, denoted as

˜

A

R

A

, is de-

ﬁned through

R

A

:

=

\

S∈R

(A ,R)

S,

˜

A

R

A

∼ A

R

, (42)

where R

(A ,R)

is the set of all those ﬁnite regions

S ⊆ G for which there exists B ∈ [[A

S

→ A

S

]]

R

such that B

S

∼ A

R

.

Lemma 23. The minimal representative exists

and is unique.

The proof follows step by step that of

lemma 73.

The ﬁrst result that we need to prove is the

following.

Lemma 24. Let A

R

∼ B

S

. Then for every ef-

fect a

T

∈ Pre[[

¯

A

G

]]

LR

, one has

A

†

R

a

T

∼ B

†

S

a

T

. (43)

Proof. If (

˜

C )

R

C

is the minimal representative of

A

R

∼ B

S

, by Eq. (41), one has

A =

˜

C ⊗ I

R\R

C

,

B =

˜

C ⊗ I

S\R

C

.

Now, by the deﬁning equation 40, we have that

(A

†

R

a

T

| = (˜a ⊗ e

(T ∪R)\R

a

|(

˜

C ⊗ I

(T ∪R)\R

C

)

=(˜a ⊗ e

[T ∪(R∩S)]\R

a

|(

˜

C ⊗ I

[T ∪(R∩S)]\R

C

)

⊗ (e

R\S

|,

(B

†

S

a

T

| = (˜a ⊗ e

(T ∪S)\R

a

|(

˜

C ⊗ I

(T ∪S)\R

C

)

=(˜a ⊗ e

[T ∪(R∩S)]\R

a

|(

˜

C ⊗ I

[T ∪(R∩S)]\R

C

)

⊗ (e

S\R

|,

and ﬁnally by Eq. (123) this implies the thesis.

By virtue of lemmas 21 and 24, we can deﬁne

the action of A

R

∈ [[A

G

→ A

G

]]

LR

on local eﬀects

[[

¯

A

G

]]

R

as

A

†

R

[a

T

]

:

= [A

†

R

a

T

],

where we leave the square braces to denote equiv-

alence classes for ∼ in Pre[[

¯

A

G

]]

LR

, for the sake of

clarity. We can now make local transformations

into a vector space as we did for local eﬀects, as

follows.

Deﬁnition 22. Let A

R

, B

S

∈ [[A

G

→ A

G

]]

LR

,

and h ∈ R. Then we deﬁne

hA

R

:

=

(

(hA )

R

h 6= 0,

0

∅

h = 0,

A

R

+ B

S

:

= C

R∪S

C

:

= A ⊗ I

S\R

+ B ⊗ I

R\S

.

Moreover, the following operation makes the

set of local transformations into an algebra.

Deﬁnition 23. Let A

R

, B

S

∈ [[A

G

→ A

G

]]

LR

.

Then we deﬁne

A

R

B

S

:

= ({A ⊗ I

S\R

}{B ⊗ I

R\S

})

R∪S

.

We omit the straightforward proof that the

above deﬁnition is well deﬁned, i.e. independent

of the choice of representatives in the classes of

A

R

and B

S

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 23

Deﬁnition 24. The algebra of generalized lo-

cal transformations is the unital algebra [[A

G

→

A

G

]]

LR

of ﬁnite real combinations of local trans-

formations, with unit 1

∅

and null element 0

∅

.

In order to close the algebra of local operations

we introduce the topology given by the sup-norm,

given in the following, in analogy to the case of

eﬀects. We also discuss the interplay of the sup-

norm topology with that given by the operational

norm, that we deﬁne right away.

Deﬁnition 25. The operational norm kA

R

k

op

of

˜

A

R

A

= A

R

∈ [[A → A]]

LR

is deﬁned by the

following expression

kA

R

k

op

:

= k

˜

A k

op

. (44)

Proposition 9. Let A

R

∈ [[A → A]]

LR

. Then

kA

R

k

op

= kA k

op

.

Proof. Let A

R

∼

˜

A

R

A

, and let R = R

A

∪ R

0

,

with R

A

∩ R

0

= ∅. Then by Eq. (41) we have

A =

˜

A ⊗ I

A

R

0

,

Now, by deﬁnition of k·k

op

(see [8]) it straightfor-

wardly follows that kB ⊗ I

C

k

op

= kBk

op

. Thus,

kA

R

k

op

= k

˜

A k

op

= kA k

op

.

Unfortunately, the operational norm does not

enjoy the basic property that would make the al-

gebra of transformations into a Banach algebra,

i.e. it is not true that kA Bk

op

≤ kA k

op

kBk

op

.

For this reason, in analogy with the case of quasi-

local eﬀects, we introduce a second norm, the sup-

norm, whose interplay with the operational norm

will make it possible to deﬁne the closed Banach

algebra of quasi-local transformations.

This is obtained extending the deﬁnition of

sup-norm to the algebra of local transformations

of A

G

.

Deﬁnition 26. The sup-norm kA

R

k

sup

of A

R

∈

[[A

G

→ A

G

]]

LR

is deﬁned as

kA

R

k

sup

:

= inf J(

˜

A ) = k

˜

A k

sup

. (45)

Proposition 10. Let A

R

∈ [[A

G

→ A

G

]]

LR

.

Then kA

R

k

sup

= kA k

sup

.

Proof. Let A

R

∼

˜

A

R

A

, and let R = R

A

∪ R

0

,

with R

A

∩ R

0

= ∅. Then by Eq. (41) we have

A =

˜

A ⊗ I

A

R

0

,

and by corollary 2, kA k

sup

= k

˜

A k

sup

=

kA

R

k

sup

.

Corollary 10. Let A , B ∈ [[A

G

→ A

G

]]

LR

.

Then kA Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. The result is a straightforward conse-

quence of propositions 10 and 2.

Corollary 11. On [[A

G

→ A

G

]]

LR

, one has

kA

R

k

op

≤ kA

R

k

sup

.

Proof. The result is a straightforward conse-

quence of propositions 9 and 10, and corol-

lary 4.

The above results allow us to extend the lo-

cal algebra of events into the quasi-local alge-

bra, which is a Banach algebra. The construc-

tion is analogous to that of quasi-local states, and

proceeds as follows. First we deﬁne the algebra

[[A

G

→ A

G

]]

CR

of sup-Cauchy sequences of local

transformations, i.e. its elements are sequences

A : N → [[A

G

→ A

G

]]

LR

:: n 7→ A

n

R

n

such that

for every ε > 0 there exists n

0

∈ N such that

for all m, n ≥ n

0

, one has kA

n

− A

m

k

sup

< ε.

Now we deﬁne the equivalence relation between

elements of [[A

G

→ A

G

]]

CR

: A

∼

=

B if for every

ε > 0 there exists n

0

∈ N such that for every

n ≥ n

0

one has kA

n

− B

n

k

sup

< ε. Finally, we

deﬁne the quasi-local algebra as follows.

Deﬁnition 27. The space of quasi-local trans-

formations of A is deﬁned as

[[A

G

→ A

G

]]

QR

:

= [[A

G

→ A

G

]]

CR

/

∼

=

.

Deﬁnition 28. An element A in [[A

G

→ A

G

]]

QR

is an event if A = [A

nR

n

] for a sequence A

nR

n

such that A

n

∈ [[A

R

n

→ A

R

n

]] for all n ∈ N. The

set of events will be denoted by [[A

G

→ A

G

]]

Q

.

An element A ∈ [[A

G

→ A

G

]]

Q

is a channel

if A = [A

n

R

n

] for a sequence A

n

R

n

such that

A

n

∈ [[A

R

n

→ A

R

n

]]

1

for all n ∈ N. The set of

channels will be denoted by [[A

G

→ A

G

]]

Q1

.

The subset of [[A

G

→ A

G

]]

QR

containing ele-

ments A = [A

n

R

n

] for a sequence A

n

R

n

such

that A

n

∈ [[A

R

n

→ A

R

n

]]

+

for all n ∈ N will be

denoted by [[A

G

→ A

G

]]

Q+

. Equivalently, we will

write A  0 for A ∈ [[A

G

→ A

G

]]

Q+

In the following, in analogy with the ﬁnite case,

we will write A  B if A − B  0. The algebra

of quasi-local transformations [[A

G

→ A

G

]]

QR

is

deﬁned as follows.

Deﬁnition 29. Let A = [A

n

R

n

], B = [B

n

S

n

].

We deﬁne A B

:

= [A

n

R

n

B

n

S

n

].

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 24

The product A B is well deﬁned. Indeed:

1. A

n

R

n

B

n

S

n

is a Cauchy sequence.

2. For every A

0

n

R

0

n

, B

0

n

S

0

n

in the equivalence

classes deﬁning A and B, respectively, one

has A

n

R

n

B

n

S

n

∼

=

A

0

n

R

0

n

B

0

n

S

0

n

.

Item 1 can be easily proved since we have

kA

n

B

n

− A

m

B

m

k

sup

=kA

n

B

n

− A

m

B

n

+ A

m

B

n

− A

m

B

m

k

sup

≤kA

n

− A

m

k

sup

kB

n

k

sup

+ kA

m

k

sup

kB

n

− B

m

k

sup

.

Similarly, for item 2 we have

kA

n

B

n

− A

0

n

B

0

n

k

sup

=kA

n

B

n

− A

0

n

B

n

+ A

0

n

B

n

− A

0

n

B

0

n

k

sup

≤kA

n

− A

0

n

k

sup

kB

n

k

sup

+ kA

0

n

k

sup

kB

n

− B

0

n

k

sup

.

The quasi-local algebra is actually a real Ba-

nach algebra if equipped with the norm k·k

sup

, as

we now prove.

Proposition 11. The algebra [[A

G

→ A

G

]]

QR

equipped with the norm k·k

sup

is a Banach al-

gebra.

Proof. We only need to prove that kA Bk

sup

≤

kA k

sup

kBk

sup

. Let A = [A

n

R

n

] and B =

[B

n

S

n

], respectively. Now, for every A

n

R

n

and

B

n

S

n

one has

k(A

n

⊗ I

S

n

\R

n

)(B

n

⊗ I

R

n

\S

n

)k

sup

≤ kA

n

⊗ I

S

n

\R

n

k

sup

kB

n

⊗ I

R

n

\S

n

k

sup

,

and by proposition 10 kA Bk

sup

≤

kA k

sup

kBk

sup

.

The subsets [[A

G

→ A

G

]]

Q

and [[A

G

→ A

G

]]

Q1

are convex, and [[A

G

→ A

G

]]

Q+

is a convex cone.

Moreover, they are closed in the sup-norm.

Proposition 12. The sets [[A

G

→ A

G

]]

Q

,

[[A

G

→ A

G

]]

Q1

, and [[A

G

→ A

G

]]

Q+

are closed

in the sup-norm.

Proof. The argument is the same in both cases.

Let {A

m

}

m∈N

be a Cauchy sequence with

A

m

∈ [[A

G

→ A

G

]]

Q∗

for all m, with ∗ =

“nothing”, 1, +. By deﬁnition, A

m

= [A

mnR

mn

],

with A

mn

∈ [[A

R

mn

→ A

R

mn

]]

∗

. Then one eas-

ily proves that the sequence {A

mmR

mm

}

m∈N

is

Cauchy, and its limit is A

:

= lim

m→∞

A

mmR

mm

,

which by deﬁnition is in [[A

G

→ A

G

]]

Q∗

.

Proposition 13. Let A ∈ [[A

G

→ A

G

]]

QR

. Then

kA k

op

≤ kA k

sup

. (46)

Proof. By deﬁnition kA k

op

= lim

n→∞

kA

n

k

op

and kA k

sup

= lim

n→∞

kA

n

k

sup

, where

{A

n

R

n

}

n∈N

⊆ [[A

G

→ A

G

]]

LR

is a Cauchy

sequence converging to A , thus by proposition 3

one has

kA

n

k

op

≤ kA

n

k

sup

, ∀n ∈ N,

which implies the inequality in the limit.

Now, we can prove the following lemma.

Lemma 25. Let a ∈ [[

¯

A

G

]]

CR

be a Cauchy se-

quence, and let A ∈ [[A

G

→ A

G

]]

LR

. We then

have

A

†

a ∈ [[

¯

A

G

]]

CR

. (47)

Proof. We just need to evaluate

kA

†

a

m

− A

†

a

n

k

sup

, and using proposition 2 one

has

kA

†

(a

m

− a

n

)k

sup

≤ ka

m

− a

n

k

sup

kA k

sup

.

Since the sequence a is Cauchy, also A

†

a is

Cauchy.

Lemma 26. Let a, b ∈ [[

¯

A

G

]]

CR

be equivalent

Cauchy sequences, and let A ∈ [[A

G

→ A

G

]]

LR

.

We then have

[A

†

a

nR

n

] = [A

†

b

nS

n

]. (48)

Proof. We can bound kA

†

a

n

− A

†

b

n

k

sup

using

proposition 2, obtaining

kA

†

(a

n

− b

n

)k

sup

≤ ka

n

− b

n

k

sup

kA k

sup

.

Since [a

nR

n

] = [b

nS

n

], the thesis follows.

As a consequence of the above results, all local

transformations leave the space [[

¯

A

G

]]

QR

invari-

ant. We now prove a further important result:

the above statement can be extended to all the

quasi-local algebra.

Theorem 7. The quasi-local algebra [[A

G

→

A

G

]]

QR

leaves the space [[

¯

A

G

]]

QR

invariant, and

for every A ∈ [[A

G

→ A

G

]]

QR

and every a ∈

[[

¯

A

G

]]

QR

, it is

kA

†

ak

sup

≤ kak

sup

kA k

sup

. (49)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 25

Proof. By proposition 2 and lemma 25, the thesis

is true for A ∈ [[A

G

→ A

G

]]

LR

and a ∈ [[

¯

A

G

]]

LR

.

Let now a = [a

pR

p

] and A ∈ [[A

G

→ A

G

]]

LR

.

Then

kA

†

a

p

k

sup

≤ ka

p

k

sup

kA k

sup

,

and taking the limit for p → ∞ we obtain the

thesis. Finally, let A = [A

nR

n

]. In this case,

by lemma 25 A

†

n

a ∈ [[

¯

A

G

]]

QR

for every n ∈ N.

Moreover, {A

†

n

a}

n∈N

∈ [[

¯

A

G

]]

CR

. Indeed, by the

result we just proved

k(A

†

n

− A

†

m

)ak

sup

≤ kak

sup

kA

n

− A

m

k

sup

.

Finally, by lemma 25, we have

kA

†

n

ak

sup

≤ kak

sup

kA

n

k

sup

,

and taking the limit for n → ∞ we obtain the

thesis.

A remarkable result that will be important in

the following is given by the following lemma.

Lemma 27. Let A ∈ [[A

G

→ A

G

]]

Q

. Then for

every a ∈ [[

¯

A

G

]]

Q

one has A

†

a ∈ [[

¯

A

G

]]

Q

.

Proof. By hypothesis, A = [A

nR

n

] and a =

[a

mS

m

], with A

n

∈ [[A

R

n

→ A

R

n

]] and a

m

∈

[[

¯

A

S

m

]]. Thus A

†

n

a

n

∈ [[

¯

A

R

n

∪S

n

]]. Now, one can

easily verify that lim

n→∞

A

†

n

a

n

= A

†

a, which

proves the statement.

In particular, one has the following condition.

Lemma 28. Let A ∈ [[A

G

→ A

G

]]

Q+

. Then

A ∈ [[A

G

→ A

G

]]

Q

iﬀ A

†

e

G

∈ [[

¯

A

G

]]

Q

.

Proof. By deﬁnition, one has A ∈ [[A

G

→ A

G

]]

Q

if and only if A = [A

nR

n

] with A

n

∈ [[A

R

n

→

A

R

n

]], and by our assumptions the latter is equiv-

alent to a

n

:

= A

†

n

e

A

R

n

∈ [[

¯

A

R

n

]]. Now, since

ka

nR

n

− a

mR

m

k

sup

≤ kA

nR

n

− A

mR

m

k

sup

, one

has that A ∈ [[A

G

→ A

G

]]

Q

if and only if

A

†

e

G

= [a

nR

n

] ∈ [[

¯

A

G

]]

Q

.

Lemma 29. Let A ∈ [[A → B]]

Q+

, and a

:

=

A

†

e

G

. Then kA k

sup

= kak

sup

.

Proof. By deﬁnition, A = [A

nR

n

], with A

n

∈

[[A

R

n

→ A

R

n

]]

+

. Now, kA

n

R

n

k

sup

= kA

n

k

sup

,

and by lemma 7, kA

n

k

sup

= kA

†

n

e

R

n

k

sup

=

kA

†

nR

n

e

G

k

sup

. Then,

kA k

sup

= lim

n→∞

kA

†

n

R

n

e

G

k

sup

= kA

†

e

G

k

sup

.

On the same line, we can provide the following

condition for A ∈ [[A

G

→ A

G

]]

Q+

to be a channel.

Lemma 30. Let A ∈ [[A

G

→ A

G

]]

Q+

. Then

A ∈ [[A

G

→ A

G

]]

Q1

iﬀ A

†

e

G

= e

G

.

Proof. First, let A be a channel. By deﬁnition

it must be A = lim

n→∞

A

nR

n

, where A

nR

n

are

local channels.

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